# Part 3: Applications of Integrals

## 3.1 Area Between Curves

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 5.6

### 3.1.1 Areas between Functions of $$x$$

• Recall that $$\int_a^b f(x)\,dx$$ is the net area between $$y=f(x)$$ and $$y=0$$.
• Let $$f(x)\leq g(x)$$ for $$a\leq x\leq b$$. We define the area between the curves $$y=f(x)$$ and $$y=g(x)$$ from $$a$$ to $$b$$ to be the integral $$\int_a^b [g(x)-f(x)]\,dx$$.
• We call $$y=f(x)$$ the bottom curve and $$y=g(x)$$ the top curve.
• Example Find the area between the curves $$y=2+x$$ and $$y=1-\frac{1}{2}x$$ from $$2$$ to $$4$$.
• Example Find the area bounded by the curves $$y=x^2-4$$ and $$y=8-2x^2$$.
• Example Prove that the area of a circle of radius $$r$$ is $$\pi r^2$$. (Hint: use the curves $$y=\pm\sqrt{r^2-x^2}$$.)

### 3.1.2 Areas between Functions of $$y$$

• Areas between functions $$f(y)\leq g(y)$$ may be found similarly, but in this case $$x=f(y)$$ is the left curve and $$x=g(y)$$ is the right curve.
• Example Find the area bounded by the curves $$y=\sqrt{x}$$, $$y=0$$, and $$y=x-2$$.

### Exercises for 3.1

1. Find the area between the curves $$y=4$$ and $$y=4x^3$$ from $$-1$$ to $$1$$.
2. Find the area bounded by the curves $$y=x^2-2x$$ and $$y=x$$.
3. Find the area bounded by the curves $$y=\pm\sqrt{4-x}$$ and $$x=3$$.
4. Find the area bounded by the curves $$y=0$$, $$x=0$$, $$y=1$$, and $$y=\ln x$$.
5. Use a definite integral to prove that the area of the triangle with vertices $$(0,0)$$, $$(b,0)$$, $$(0,h)$$ is $$\frac{1}{2}bh$$.
6. (Optional) Find the area of the ellipse $$9x^2+16y^2=25$$.

Solutions

## 3.2 Volumes by Cross-Sectioning

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 6.1

### 3.2.1 Defining Volume with Integrals

• The volume of a solid defined between $$x=a$$ to $$x=b$$ with a cross-sectional area of $$A(x)$$ at each $$x$$-value is defined to be $$V=\int_a^b A(x)\,dx$$.
• Steps for solving such problems:
1. Sketch the solid along the $$x$$-axis with a typical cross-section at some $$x$$ value.
2. Find the formula for $$A(x)$$, and the minimal/maximal $$x$$ values $$a,b$$.
3. Evaluate $$V=\int_a^b A(x)\,dx$$.
• Example Show that the volume of a pyramid with a square base of sidelength $$2$$ and height $$3$$ is $$4$$ cubic units.

### 3.2.2 Circular Cross-Sections

• In the case that all cross-sections are circular, we may replace $$A(x)$$ with $$\pi[R(x)]^2$$, where $$R(x)$$ is the radius of the circular cross-section at that $$x$$ value.
• Example Prove that a cone of radius $$r$$ and height $$h$$ has volume $$V=\frac{1}{3}\pi r^2 h$$.

### Exercises for 3.2

1. Find the volume of a solid located between $$x=-1$$ and $$x=2$$ with cross-sectional area $$A(x)=x^2+1$$ for all $$-1\leq x\leq 2$$.
2. Find the volume of a solid located between $$x=0$$ and $$x=1$$ whose cross-sections are parallelograms with base length $$b(x)=x+1$$ and height $$h(x)=x^2+1$$ for all $$0\leq x\leq 1$$.
3. Find the volume of a wedge cut from a circular cylinder with radius $$2$$, sliced out at a $$45^\circ$$ angle from the diameter of its base. (Hint: Sketch the diameter of the cylinder along the $$x$$-axis from $$-2$$ to $$2$$, and use the equation $$x^2+y^2=2^2$$. The cross-sections will be isosceles triangles.)
4. Prove that the volume of a sphere with radius $$r$$ is $$V=\frac{4}{3}\pi r^3$$. (Hint: Draw a diameter of the sphere on the $$x$$-axis from $$-r$$ to $$r$$, and use the equation $$x^2+y^2=r^2$$.)
5. (Optional) Find the volume of the solid whose base is the region $$0\leq y\leq 4-x^2$$ and whose cross-sections are equilateral triangles perpendicular to the $$x$$-axis.

Solutions

## 3.3 The Washer Method

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 6.1

Lecture Notes

### 3.3.1 Rotation about Horizontal Axes

• Many solids may be described as revolutions of two-dimensional regions. Such solids have washer-shaped cross-sections.
• To obtain the volume of a solid of revolution about a horizontal axis, identify the outer radius $$R(x)$$ and inner radius $$r(x)$$ for each $$x$$-value, the leftmost $$a$$ and rightmost $$b$$ $$x$$-values in the region and use the formula $$V=\int_a^b \pi([R(x)]^2-[r(x)]^2)\,dx$$.
• Example Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(0,0)$$, $$(2,2)$$, $$(2,4)$$ around the $$x$$-axis.
• Example Find the volume of the solid of revolution obtained by rotating the region bounded by $$y=x$$ and $$y=x^2$$ around the axis $$y=2$$.

### 3.3.2 Rotation about Vertical Axes

• When the axis of revolution is vertical, simply use functions of $$y$$ rather than $$x$$, and the bottommost $$c$$ and topmost $$d$$ $$y$$-values: $$V=\int_c^d \pi([R(y)]^2-[r(y)]^2)\,dy$$.
• Example Find the volume of the solid of revolution obtained by rotating the region bounded by $$y=0$$, $$x=1$$, $$y=\sqrt{x}$$ around the axis $$x=-1$$.

### Exercises for 3.3

1. Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(3,0)$$, $$(3,3)$$, $$(0,3)$$ around the $$x$$-axis.
2. Find the volume of the solid of revolution obtained by rotating the region bounded by $$y=x^2$$, $$y=2x$$ around the axis $$y=-2$$.
3. Consider the region in the $$xy$$ plane satisfying $$|x|\leq\frac{\pi}{2}$$ and $$|y|\leq\cos x$$. Find the volume of the solid of revolution obtained by rotating this region around the axis $$y=3$$.
4. Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(0,0)$$, $$(2,0)$$, $$(2,1)$$ around the axis $$x=4$$.
5. Find the volume of the solid of revolution obtained by rotating the region bounded by $$x+y=1$$, $$y=\ln x$$, $$y=1$$ around the $$y$$-axis.
6. (Optional) Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(-\sqrt 2,0)$$, $$(0,\sqrt 2)$$, $$(\sqrt 2,0)$$ around the axis $$y=\sqrt 2-x$$. (Hint: Translate the region and its axis so that it has a horizontal or vertical axis of revolution.)
7. (Quiz, 3.1 material) Find the area between the curves $$y=x^2$$ and $$y=4$$.
• $$\frac{32}{3}$$
• $$\frac{25}{4}$$
• $$7$$
8. (Quiz) What integral is produced by the washer method for the volume of the solid of revolution obtained by rotating the region bounded by $$y=x^2$$ and $$y=4$$ around the $$x$$-axis?
• $$\pi\int_{-2}^2[(4)^2-(x^2)^2]\,dx$$
• $$\pi\int_0^2(x^2-4)^2\,dx$$
• $$\int_{-1}^1 2\sqrt{\pi}-y\sqrt{\pi}\,dy$$
9. (Quiz) What integral is produced by the washer method for the volume of the solid of revolution obtained by rotating the triangle with vertices $$(1,1)$$, $$(2,1)$$, $$(2,0)$$ around the axis $$x=3$$?
• $$\pi\int_0^1[(1+y)^2-(1)^2]\,dy$$
• $$\int_1^2[\pi(2-y)^2-(3)^2]\,dy$$
• $$\pi\int_1^3[(2)^2-(2+x)^2]\,dx$$

Solutions 1-3

Solutions 4-6

Solutions 7-9

## 3.4 The Cylindrical Shell Method

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 6.2

### 3.4.1 Rotation about Vertical Axes

• As an alternative to the washer method, one may consider “cylindrical shell” cross-sections instead.
• The lateral surface area of a cylinder is given by $$2\pi rh$$.
• For a vertical axis of revolution, identify the radius $$r(x)$$ and height $$h(x)$$ of a cylindrical shell, identify the leftmost $$a$$ and rightmost $$b$$ $$x$$-values of the region, and use the formula $$V=\int_a^b 2\pi r(x)h(x)\,dx$$.
• Example Find the volume of the solid of revolution obtained by rotating the region bounded by $$y=0$$, $$x=1$$, $$y=\sqrt{x}$$ around the line $$x=-1$$.
• Example Find the volume of the solid of revolution obtained by rotating the region bounded by $$y=0$$, $$x=\pm1$$, $$y=x^2+1$$ around the line $$x=2$$.

### 3.4.2 Rotation about Horizontal Axes

• When the axis of revolution is horiztonal, simply use functions of $$y$$ rather than $$x$$, and the bottommost $$c$$ and topmost $$d$$ $$y$$-values: $$V=\int_c^d 2\pi r(y)h(y)\,dy$$.
• Example Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(-1,2)$$, $$(0,1)$$, $$(2,2)$$ around the $$x$$-axis.

### Exercises for 3.4

1. Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(0,2)$$, $$(1,0)$$, $$(1,2)$$ around the axis $$x=2$$.
2. Find the volume of the solid of revolution obtained by rotating the region bounded by $$y=4$$, $$y=x^2-4x+4$$ around the $$y$$-axis.
3. Find the volume of the solid of revolution obtained by rotating the region bounded by $$x=y^2-1$$, $$x=3$$ around the axis $$x=-1$$.
4. Find the volume of the solid of revolution obtained by rotating the triangle with vertices $$(4,2)$$, $$(2,6)$$, $$(0,6)$$ around the axis $$y=2$$.
5. Find the volume of the solid of revolution obtained by rotating the region bounded by $$x=e$$, $$y=2$$, $$y=\ln x$$ around the $$x$$-axis.
6. (Optional) Use the cylindrical shell method to reprove the volume formula for a sphere: $$V=\frac{4}{3}\pi R^3$$.
7. (Quiz) What integral is produced by the cylindrical shell method for the volume of the solid of revolution obtained by rotating the triangle with vertices $$(0,0),(2,0),(0,4)$$ around the $$y$$-axis?
• $$\pi\int_0^2(2x^2)(2x+4)\,dx$$
• $$\int_{-4}^4(2\pi-y)\,dy$$
• $$2\pi\int_0^2(x)(4-2x)\,dx$$
8. (Quiz) What integral is produced by the cylindrical shell method for the volume of the solid of revolution obtained by rotating the region bounded by $$x=0,y=2,x=y^3$$ around the axis $$y=-1$$?
• $$2\pi\int_0^2(y+1)(y^3),dy$$
• $$2\pi\int_8^0(y-1)(y^3+1)\,dy$$
• $$2\pi\int_1^3(2x)^2(\sqrt[3]x)\,dx$$

Solutions 1-3

Solutions 4-8

## 3.5 Work

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 6.5

### 3.5.1 Work by a Constant Force

• In physics, the work $$W$$ done by a force of constant magnitude $$F$$ over a displacement $$d$$ by the formula $$W=Fd$$.
• Example Calculate the work done by a crane in lifting a $$3000$$ pound wrecking ball $$25$$ feet.
• Example Estimate the work done in lifting a leaky bucket of water $$1$$ meter off the ground if it weighs approximately $$4$$ newtons on the ground, $$3.8$$ newtons at $$25$$ cm, $$3.5$$ newtons at $$50$$ cm, and $$2.3$$ newtons at $$75$$ cm.

### 3.5.2 Work by a Variable Force

• If the force $$F(x)$$ acting on an object varies with respect to the position $$x$$ of the object, then work done in moving the object from $$a$$ to $$b$$ is defined by $$W=\int_a^b F(x)\,dx$$.
• Example Find the work done in lifting a leaky bucket of water $$1$$ meter off the ground if it weighs $$4-3x^2$$ newtons when it is $$x$$ meters above the ground.
• Example How much work is done in pulling up $$20$$ feet of hanging chain if it weighs $$1$$ pound per $$4$$ feet?
• Example Hooke’s Law states that the force required to hold a stretched or compressed spring is directly proportional to its natural length. That is, $$F(x)=kx$$ where $$x$$ is the difference between the spring’s natural length and its current length. If a spring has natural length $$10$$ inches, and it requires $$15$$ pounds of force to hold the spring at $$13$$ inches, how much work is required to stretch the spring an additional $$2$$ inches?

### 3.5.3 Work and Pumping Liquid

• To compute the work in pumping liquid, we proceed by computing a work differential $$dW$$ for each infintesimal cross-section of liquid at height $$y$$, and then evaluating $$W=\int_{y=a}^{y=b} dW$$ where $$y=a$$ is the lowest point of liquid and $$y=b$$ is the highest.
• Example Assume salt water weighs $$10,000$$ newtons per cubic meter. How much work is required to pump out a conical tank pointed downward of height $$6$$ meters and radius $$3$$ meters, if it is initially filled with $$4$$ feet of salt water?

### Exercises for 3.5

1. Estimate the work done in pushing a plow $$6$$ meters through increasingly packed dirt; this movement requires $$1$$ newton of force at the beginning, $$5$$ newtons of force after $$2$$ meters, and $$9$$ netwons of force after $$4$$ meters.
2. Compute the exact amount of work done in pushing a plow $$6$$ meters through increasingly packed dirt; this movement requires $$F(x)=1+2x$$ newtons of force after $$x$$ meters.
3. Find the work done in lifting a leaky bucket from the ground to a height of four feet, assuming it weighs $$25-x$$ pounds at $$x$$ feet above the ground.
4. A cable weighing $$4$$ pounds per foot holds a $$500$$ pound bucket of coal at the bottom of a $$300$$ foot mine shaft. Show that the total work done in lifting the cable and bucket is $$330,000$$ foot-pounds.
5. Hooke’s Law states that the force required to hold a stretched or compressed spring is directly proportional to its natural length. That is, $$F(x)=kx$$ where $$x$$ is the difference between the spring’s natural length and its current length. Show that if a spring has natural length $$20$$ cm, and it requires $$25$$ newtons of force to hold the spring at $$15$$ cm, then the work required to stretch the spring from its natural length to $$26$$ cm is $$90$$ N-cm.
6. A uniformly weighted $$100$$ foot rope weighs $$50$$ pounds. Suppose it is fully extended into a well, tied to a leaky bucket of water. This bucket weighs $$10$$ pounds and initially holds $$30$$ pounds of water, but loses $$1$$ pound of water every $$2$$ feet. Show that the work done in lifting the rope and bucket is $$4400$$ ft-lbs. (Hint: When does the bucket run out of water?)
7. Assume salt water weighs $$10$$ kilonewtons (kN) per cubic meter. A cylindrical tank with a radius of $$3$$ meters and a height of $$10$$ meters holds $$8$$ meters of salt water. Show that the work required to pump out the salt water to the top of the tank is $$4320\pi$$ kN-m (kJ).
8. Assume salt water weighs $$10000$$ newtons per cubic meter. A pyramid-shaped tank of height $$4$$ meters is pointed upward, with a square base of side length $$4$$ meters, and is completely filled with salt water. Show that the work done in completely pumping all the water in the tank up to the point of the pyramid is $$10000\int_0^4(4-y)^3\,dy$$ J.
9. (Optional) Assume that a cubic inch of Juicy Juice(TM) weighs $$D$$ oz. Suppose a perfectly spherical coconut-shaped cup with radius $$R$$ inches is completely filled with Juicy Juice(TM). Show that drinking the entire beverage using a straw which extends $$S$$ inches above the top of the container requires $$\frac{4}{3}D\pi R^3(R+S)$$ inch-ounces of work.
10. What is the work required to push a heavy box $$3$$ meters over an irregular surface, assuming it requires $$F(x)=3+2x-x^2$$ newtons of force to move at $$x$$ meters?
• $$\frac{5}{3}$$ joules
• $$9$$ joules
• $$\frac{13}{3}$$ joules
11. Which of these integrals gives the work in ft-lbs required to pull up a hanging $$30$$-pound $$15$$-foot chain?
• $$\int_0^{15}(60-2x)\,dx$$
• $$\int_0^{15x}30\,dy$$
• $$\int_{15}^{30}(15+x)\,dx$$
12. Which of these integrals gives the work in kN-m required to pump out all salt-water to the top of a cubical tank with side length $$4$$ meters, if it is initially half-full? Assume the density of salt water is $$10$$ kilonewtons per cubic meter.
• $$10\pi\int_0^4(16+8y+y^2)\,dy$$
• $$\int_0^{16}(4-y)^3\,dy$$
• $$160\int_0^2(4-y)\,dy$$