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

- 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 \(y=\pm\sqrt{r^2-x^2}\).)

- 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 \(y=x-2\).

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

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

- 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:
- Sketch the solid along the \(x\)-axis with a typical cross-section at some \(x\) value.
- Find the formula for \(A(x)\), and the minimal/maximal \(x\) values \(a,b\).
- 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.

- 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 \(V=\frac{1}{3}\pi r^2 h\).

- Find the volume of a solid located between \(x=-1\) and \(x=2\) with cross-sectional area \(A(x)=x^2+1\) for all \(-1\leq x\leq 2\).
- 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 \(0\leq x\leq 1\).
- 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.)
- 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 \(x^2+y^2=r^2\).)
- (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.

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

- 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 \(y=2\).

- 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 \(x=-1\).

- 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.
- Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=x^2\), \(y=2x\) around the axis \(y=-2\).
- 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 \(y=3\).
- Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,0)\), \((2,0)\), \((2,1)\) around the axis \(x=4\).
- 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.
- (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.)
- (Quiz, 3.1 material)
Find the area between the curves \(y=x^2\) and \(y=4\).
- \(\frac{32}{3}\)
- \(\frac{25}{4}\)
- \(7\)

- (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?
- \(\pi\int_{-2}^2[(4)^2-(x^2)^2]\,dx\)
- \(\pi\int_0^2(x^2-4)^2\,dx\)
- \(\int_{-1}^1 2\sqrt{\pi}-y\sqrt{\pi}\,dy\)

- (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 \(x=3\)?
- \(\pi\int_0^1[(1+y)^2-(1)^2]\,dy\)
- \(\int_1^2[\pi(2-y)^2-(3)^2]\,dy\)
- \(\pi\int_1^3[(2)^2-(2+x)^2]\,dx\)

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

- 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 \(x=2\).

- 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.

- Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,2)\), \((1,0)\), \((1,2)\) around the axis \(x=2\).
- 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.
- Find the volume of the solid of revolution obtained by rotating the region bounded by \(x=y^2-1\), \(x=3\) around the axis \(x=-1\).
- Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((4,2)\), \((2,6)\), \((0,6)\) around the axis \(y=2\).
- 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.
- (Optional) Use the cylindrical shell method to reprove the volume formula for a sphere: \(V=\frac{4}{3}\pi R^3\).
- (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?
- \(\pi\int_0^2(2x^2)(2x+4)\,dx\)
- \(\int_{-4}^4(2\pi-y)\,dy\)
- \(2\pi\int_0^2(x)(4-2x)\,dx\)

- (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 \(y=-1\)?
- \(2\pi\int_0^2(y+1)(y^3),dy\)
- \(2\pi\int_8^0(y-1)(y^3+1)\,dy\)
- \(2\pi\int_1^3(2x)^2(\sqrt[3]x)\,dx\)

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

- 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.

- 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?

- 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?

- 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.
- 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.
- 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.
- 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.
- 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.
- 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?)
- 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).
- 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.
- (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.
- 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

- Which of these integrals gives the work in ft-lbs
required to pull up a hanging \(30\)-pound \(15\)-foot chain?
- \(\int_0^{15}(60-2x)\,dx\)
- \(\int_0^{15x}30\,dy\)
- \(\int_{15}^{30}(15+x)\,dx\)

- 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.
- \(10\pi\int_0^4(16+8y+y^2)\,dy\)
- \(\int_0^{16}(4-y)^3\,dy\)
- \(160\int_0^2(4-y)\,dy\)