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

1 Checking an antiderivative ★★★

Show that \(F(x)=\dfrac{x^4}{4}-3x\) is an antiderivative of \(f(x)=x^3-3\).

2 A general antiderivative ★★★

Find the general antiderivative of \(f(x)=8x^3-6x^2+2x-7\).

3 A first definite integral ★★★

Evaluate \(\displaystyle\int_1^4 (2x+3)\,dx\).

4 Using the properties ★★★

Suppose \(\displaystyle\int_1^5 f(x)\,dx=7\) and \(\displaystyle\int_1^5 g(x)\,dx=-2\). Find:

1. \(\displaystyle\int_1^5 \big(2f(x)+5g(x)\big)dx\)
2. \(\displaystyle\int_5^1 f(x)\,dx\)
3. \(\displaystyle\int_1^5 \big(f(x)-g(x)\big)dx\)

5 Left and right sums ★★★

For \(f(x)=2x\) on \([0,3]\) with \(n=3\) strips, compute the left sum and the right sum. Then compare them with the exact area under the line.

6 Fixing the constant ★★★

Find the function \(F\) with \(F'(x)=6x+1\) and \(F(2)=10\).

7 True or false? ★★★

Decide whether each statement is true or false, and justify.

1. \(\displaystyle\int_{-2}^{2} x^3\,dx=0\).
2. \(\displaystyle\int_0^2 \big(1\cdot 1\big)dx=\left(\int_0^2 1\,dx\right)\left(\int_0^2 1\,dx\right)\).

8 Sine and the natural log ★★★

Evaluate:

1. \(\displaystyle\int_0^{\pi}\sin x\,dx\)
2. \(\displaystyle\int_1^{e}\dfrac1x\,dx\)

9 A midpoint estimate ★★★

Use the midpoint rule with \(n=4\) strips to approximate \(\displaystyle\int_0^4 (x^2+1)\,dx\). Then compute the exact value and the error.

10 Area of a tunnel opening ★★★

The cross-section of a tunnel is the region between the x-axis and the parabola \(y=9-x^2\), with \(x\) and \(y\) in meters. Find its area in square meters, and convert it to square feet (1 m\(^2\approx10.76\) ft\(^2\)).

11 Filling a water tank ★★★

A large tank contains 80 gallons of water at time \(t=0\). Water is pumped in at a rate of \(r(t)=12-2t\) gallons per minute for \(0\le t\le 5\) minutes. How many gallons does the tank contain after 5 minutes? Give the answer in liters too (1 gal \(\approx 3.785\) L).

12 Two quick substitutions ★★★

Find each indefinite integral.

1. \(\displaystyle\int 2x\cos(x^2)\,dx\)
2. \(\displaystyle\int (3x+2)^4\,dx\)

13 Substitution with new limits ★★★

Evaluate \(\displaystyle\int_0^1\dfrac{x}{x^2+1}\,dx\).

14 Differentiating an integral ★★★

Let \(g(x)=\displaystyle\int_2^x (t^3-5)\,dt\). Find \(g'(x)\), then \(g'(3)\) and \(g(2)\).

15 The chain rule meets the FTC ★★★

Find the derivatives:

1. \(h(x)=\displaystyle\int_0^{x^2}\sqrt{1+t^2}\,dt\); also compute \(h'(1)\).
2. \(k(x)=\displaystyle\int_x^5\sin(t^2)\,dt\).

16 A drone changes direction ★★★

A drone flies along a straight line with velocity \(v(t)=t^2-6t+8\) m/s for \(0\le t\le5\) seconds. Find its displacement and the total distance traveled.

17 Sine cubed times cosine ★★★

Evaluate \(\displaystyle\int_0^{\pi/2}\sin^3x\cos x\,dx\).

18 Logarithm substitution ★★★

Evaluate \(\displaystyle\int_1^{e^2}\dfrac{\ln x}{x}\,dx\).

19 Absolute value integral ★★★

Compute \(\displaystyle\int_{-1}^{3}|x-1|\,dx\). (Hint: split the interval where \(x-1\) changes sign.)

20 Marginal cost ★★★

A factory’s marginal cost is \(C'(q)=0.06q+4\) dollars per unit when \(q\) units have been made. By how many dollars does the total cost rise when production goes from 100 to 200 units?

21 From Riemann sums to an integral ★★★

Let \(R_n\) be the right-endpoint sum of \(f(x)=x^2\) on \([0,1]\) with \(n\) strips. Use \(\sum_{i=1}^n i^2=\dfrac{n(n+1)(2n+1)}{6}\) to find a formula for \(R_n\), compute \(R_{10}\), and find \(\lim_{n\to\infty}R_n\).

22 A radical integrand ★★★

Evaluate \(\displaystyle\int_0^2x\sqrt{x^2+5}\,dx\) exactly, then give a decimal approximation.