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

1 Area between a parabola and a line / 4 pts

Let \(y=x^2\) and \(y=2x\).

1. Find the intersection points.
2. State which curve is on top between them.
3. Compute the area enclosed.

2 Volume with disks / 4 pts

The region under \(y=e^x\) on \([0,1]\) is rotated about the x-axis. A calculator is allowed.

1. Write the volume as an integral.
2. Evaluate it exactly.
3. Give a decimal value to two places.

3 Volume with shells / 4 pts

The region under \(y=3x-x^2\) for \(0\le x\le3\) is rotated about the y-axis. Set up and compute the volume using shells.

4 Arc length / 3 pts

Find the length of \(y=\tfrac{x^3}{3}+\tfrac{1}{4x}\) for \(1\le x\le2\). Hint: show that \(1+(y')^2\) is a perfect square.

5 Average value and work / 3 pts

(a) Find the average value of \(\sin x\) on \([0,\pi]\).

(b) A force of 6 N holds a spring stretched 0.15 m. Find the work to stretch it from natural length to 0.3 m.

6 Separable equation / 2 pts

Solve \(\dfrac{dy}{dx}=2xy\) with \(y(0)=1\).