Integrals and the Fundamental Theorem: math test, College – download the PDF

1 Antiderivatives / 3 pts

1. Find the general antiderivative of \(f(x)=5x^4-3x^2+x-2\).
2. Find the antiderivative \(F\) of \(g(x)=6x^2+4x\) satisfying \(F(1)=3\).

2 Riemann sums / 4 pts

Calculator allowed. Let \(f(x)=x^2+2x\) on \([0,2]\) and \(n=4\).

1. Compute the right-endpoint sum.
2. Compute the left-endpoint sum.
3. Compute the exact integral and check that it lies between the two sums.

3 Properties / 3 pts

Suppose \(\displaystyle\int_0^3f=4\), \(\displaystyle\int_3^7f=-1\) and \(\displaystyle\int_0^7g=5\). Find:

1. \(\displaystyle\int_0^7f\)
2. \(\displaystyle\int_7^3f\)
3. \(\displaystyle\int_0^7(2f-g)\)

4 Fundamental Theorem / 4 pts

1. Evaluate \(\displaystyle\int_0^2(6x^2-2x)\,dx\).
2. If \(F(x)=\displaystyle\int_0^x\sqrt{t^2+9}\,dt\), find \(F'(4)\).
3. If \(G(x)=\displaystyle\int_0^{2x}(t^2+1)\,dt\), find \(G'(x)\) and \(G'(1)\).

5 Net change / 3 pts

A town’s water use rate is \(w(t)=200+30t\) gallons per hour, where \(t\) is the number of hours after 6 a.m. and \(0\le t\le4\).

1. How many gallons are used between 6 a.m. and 10 a.m.?
2. A reservoir holds 5,000 gallons at 6 a.m. and receives no water. How much remains at 10 a.m.?

6 Substitution / 3 pts

1. Find \(\displaystyle\int 6x(x^2-4)^3\,dx\).
2. Evaluate \(\displaystyle\int_0^1x^2e^{x^3}\,dx\).