Part 3: Applications of Integrals


3.1 Area Between Curves

Textbook References

3.1.1 Areas between Functions of \(x\)

3.1.2 Areas between Functions of \(y\)

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

3.2.1 Defining Volume with Integrals

3.2.2 Circular Cross-Sections

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

Lecture Notes

3.3.1 Rotation about Horizontal Axes

3.3.2 Rotation about Vertical Axes

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

3.4.1 Rotation about Vertical Axes

3.4.2 Rotation about Horizontal Axes

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

3.5.1 Work by a Constant Force

3.5.2 Work by a Variable Force

3.5.3 Work and Pumping Liquid

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

Solutions 1-6 Solutions 7-12