Applications of Integration: math practice, College – download the PDF

1 Area between a line and a parabola ★★★

Find the area of the region between \(y=x\) and \(y=x^2\) for \(0\le x\le 1\).

2 Area under a roof line ★★★

A decorative window is bounded above by \(y=4\) and below by \(y=x^2\), with \(x\) in feet. Find its area.

3 A cone from a line ★★★

Rotate the region under \(y=x\) for \(0\le x\le 3\) about the x-axis. Find the volume with disks and check it with the cone formula.

4 Length of a ramp ★★★

Use the arc length formula to find the length of \(y=3x+1\) from \(x=0\) to \(x=4\), then check with the distance formula.

5 Average of a linear function ★★★

Find the average value of \(f(x)=2x+1\) on \([0,4]\). At which \(x\) does \(f\) take that value?

6 Constant force ★★★

A worker pushes a crate with a constant 25 lb force over 12 ft. Find the work in foot-pounds and in joules (1 ft\(\cdot\)lb \(\approx\) 1.3558 J).

7 Checking a differential equation ★★★

Show that \(y=Ce^{-2x}\) solves \(y'=-2y\) for every constant \(C\). Then find the solution with \(y(0)=5\) and compute \(y(1)\).

8 Square root against a line ★★★

Find the area between \(y=\sqrt{x}\) and \(y=\tfrac{x}{2}\) (see the figure).

9 Curves that cross ★★★

Find the total area enclosed by \(y=x\) and \(y=x^3\).

10 Washer about the x-axis ★★★

The region between \(y=2x\) and \(y=x^2\) is rotated about the x-axis. Find the volume.

11 A cubic bowl ★★★

Rotate the region under \(y=x^3\) for \(0\le x\le1\) about the x-axis. Find the volume.

12 A vase by shells ★★★

Use shells to find the volume when the region under \(y=4x-x^2\) (for \(0\le x\le 4\)) is rotated about the y-axis.

13 Length of a curved cable ★★★

A hanging cable follows \(y=x^{3/2}\) for \(0\le x\le 4\) (feet). Find its length.

14 Stretching a spring in metric units ★★★

A spring has constant \(k=200\) N/m. How much work is needed to stretch it from 0.1 m to 0.3 m beyond its natural length?

15 Average temperature ★★★

The temperature in a greenhouse is \(T(t)=60+20\sin\!\big(\tfrac{\pi t}{12}\big)\) degrees Fahrenheit, \(t\) hours after 6 a.m., for \(0\le t\le12\). Find the average temperature over these 12 hours.

16 Integrating with respect to y ★★★

Find the area enclosed by the parabola \(x=y^2\) and the line \(x=y+6\).

17 Shells about a vertical line ★★★

The region under \(y=x^2\) on \([0,1]\) is rotated about the line \(x=2\) (dashed in the figure). Use shells to find the volume.

18 Washers about a horizontal line ★★★

The region between \(y=x\) and \(y=x^2\) on \([0,1]\) is rotated about the line \(y=-1\). Find the volume.

19 An arc that needs a table ★★★

Find the length of \(y=\tfrac{x^2}{2}\) from \(x=0\) to \(x=1\). You may use \(\int\sqrt{1+x^2}\,dx=\tfrac12\big(x\sqrt{1+x^2}+\ln(x+\sqrt{1+x^2})\big)+C\).

20 Pumping water from a tank ★★★

A cylindrical tank of radius 3 ft and height 8 ft is full of water (62.4 lb/ft\(^3\)). How much work is needed to pump all the water to the top rim?

21 A separable initial-value problem ★★★

Solve \(\dfrac{dy}{dx}=xe^{-y}\) with \(y(0)=0\), then find \(y(2)\).

22 Cooling coffee ★★★

Newton’s law says \(\dfrac{dT}{dt}=-k(T-70)\) for coffee in a 70 \(^\circ\)F room. The coffee starts at 190 \(^\circ\)F and is 150 \(^\circ\)F after 10 minutes. Find its temperature after 20 minutes and the time to reach 100 \(^\circ\)F.