Definite integrals by hand and with the calculator Problem set B

Evaluate these definite integrals without a calculator.

1. (a) \(\int_{0}^{32} x^{\frac{1}{5}}\,dx\);
2. (b) \(\int_{1}^{27} x^{- \frac{2}{3}}\,dx\).

Evaluate these definite integrals without a calculator.

1. (a) \(\int_{-4}^{1} x^{3}\,dx\);
2. (b) \(\int_{-6}^{-3} x^{2}\,dx\).

Find the exact value of each integral. Give the answers in terms of \(\pi\) or surds where needed.

1. (a) \(\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \sin x\,dx\);
2. (b) \(\int_{0}^{\frac{\pi}{3}} \cos x\,dx\).

Find the exact value of each integral.

1. (a) \(\int_{0}^{\ln 9} e^{x}\,dx\);
2. (b) \(\int_{\ln 3}^{\ln 7} 5e^{x}\,dx\).

Find the exact value of each integral.

1. (a) \(\int_{3}^{11} \frac{1}{x}\,dx\);
2. (b) \(\int_1^{e^5} \frac{2}{x}\,dx\).

Find the exact value of \(\int_0^{\pi/9}\frac{9}{\cos^2(3x)}\,dx\). Then give it to 3 significant figures.

A factory pipe releases dye into a river at a rate of \(c(t) = 14e^{-t^2/6}\) grams per second, where t is the time in seconds after a valve is opened. Use your calculator to find how much dye is released in the first 4 seconds. Give your answer to 3 significant figures.

A drone climbs from a standstill. Its speed is \(v = 3t^{2/3}\) m/s, where t is the time in seconds after take-off. Find the distance it travels in the first 8 seconds.

A video is going viral. The number of views is increasing at the rate \(V'(t) = 4e^{t}\) thousand views per day, where t is the time in days. Find the number of views added between \(t = 0\) and \(t = \ln 7\).

Given that \(\int_1^k \frac{3}{x}\,dx = \ln 64\) and \(k > 1\), find the exact value of k.

Find the exact value of \(\int_{1}^{8} \left(x^{\frac{1}{3}} + \frac{4}{x}\right)dx\), then give it to 3 significant figures.