Definite integrals by hand and with the calculator Problem set A

Evaluate these definite integrals without a calculator.

1. (a) \(\int_{1}^{16} x^{\frac{1}{4}}\,dx\);
2. (b) \(\int_{1}^{8} x^{- \frac{2}{3}}\,dx\).

Evaluate these definite integrals without a calculator.

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

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

1. (a) \(\int_{0}^{\frac{2 \pi}{3}} \sin x\,dx\);
2. (b) \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cos x\,dx\).

Find the exact value of each integral.

1. (a) \(\int_{0}^{\ln 6} e^{x}\,dx\);
2. (b) \(\int_{\ln 2}^{\ln 5} 4e^{x}\,dx\).

Find the exact value of each integral.

1. (a) \(\int_{2}^{7} \frac{1}{x}\,dx\);
2. (b) \(\int_1^{e^4} \frac{3}{x}\,dx\).

Find the exact value of \(\int_0^{\pi/12}\frac{10}{\cos^2(2x)}\,dx\). Then give it to 3 significant figures.

Water flows into a tank at a rate of \(r(t) = 9e^{-t^2/8}\) litres per minute, where t is the time in minutes after the tap is opened. Use your calculator to find how much water flows in during the first 5 minutes. Give your answer to 3 significant figures.

A cyclist accelerates away from rest. Her speed is \(v = 2t^{3/2}\) m/s, where t is the time in seconds after she starts. Find the distance she travels in the first 4 seconds.

A culture of bacteria is growing. Its size increases at the rate \(B'(t) = 6e^{t}\) thousand bacteria per hour, where t is the time in hours. Find the number of bacteria added between \(t = 0\) and \(t = \ln 4\).

Given that \(\int_1^k \frac{2}{x}\,dx = \ln 25\) and \(k > 1\), find the exact value of k.

Find the exact value of \(\int_{1}^{4} \left(x^{\frac{1}{2}} + \frac{6}{x}\right)dx\), then give it to 3 significant figures.