Definite integrals by hand and with the calculator Problem set A Worked solutions

Evaluate these definite integrals without a calculator.

1. (a) \(\int_{1}^{16} x^{\frac{1}{4}}\,dx\);

   • \(\int_{1}^{16} x^{\frac{1}{4}}\,dx = \left[\frac{4}{5}x^{\frac{5}{4}}\right]_{1}^{16}\)
   • \(= \frac{128}{5} - \frac{4}{5}\)
   • \(= \frac{124}{5}\)

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

   • \(\int_{1}^{8} x^{- \frac{2}{3}}\,dx = \left[3x^{\frac{1}{3}}\right]_{1}^{8}\)
   • \(= 6 - 3\)
   • \(= 3\)

Evaluate these definite integrals without a calculator.

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

   • \(\int_{-3}^{2} x^{3}\,dx = \left[\frac{1}{4}x^{4}\right]_{-3}^{2}\)
   • \(= 4 - \frac{81}{4}\)
   • \(= - \frac{65}{4}\)

2. (b) \(\int_{-5}^{-2} x^{2}\,dx\).

   • \(\int_{-5}^{-2} x^{2}\,dx = \left[\frac{1}{3}x^{3}\right]_{-5}^{-2}\)
   • \(= - \frac{8}{3} - \left(- \frac{125}{3}\right)\)
   • \(= 39\)

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

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

   • \(\int_{0}^{\frac{2 \pi}{3}} \sin x\,dx = \left[-\cos x\right]_{0}^{\frac{2 \pi}{3}}\)
   • \(= \left(-\cos \frac{2 \pi}{3}\right) - \left(-\cos 0\right)\)
   • \(= \frac{1}{2} - \left(-1\right)\)
   • \(= \frac{3}{2}\)

2. (b) \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cos x\,dx\).

   • \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cos x\,dx = \left[\sin x\right]_{\frac{\pi}{6}}^{\frac{\pi}{2}}\)
   • \(= \left(\sin \frac{\pi}{2}\right) - \left(\sin \frac{\pi}{6}\right)\)
   • \(= 1 - \frac{1}{2}\)
   • \(= \frac{1}{2}\)

Find the exact value of each integral.

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

   • \(\int_{0}^{\ln 6} e^{x}\,dx = \left[e^x\right]_{0}^{\ln 6}\)
   • \(= e^{\ln 6} - e^{0}\)
   • \(= 6 - 1\)
   • \(= 5\)

2. (b) \(\int_{\ln 2}^{\ln 5} 4e^{x}\,dx\).

   • \(\int_{\ln 2}^{\ln 5} 4e^{x}\,dx = \left[4e^x\right]_{\ln 2}^{\ln 5}\)
   • \(= 4e^{\ln 5} - 4e^{\ln 2}\)
   • \(= 4\times5 - 4\times2\)
   • \(= 12\)

Find the exact value of each integral.

1. (a) \(\int_{2}^{7} \frac{1}{x}\,dx\);

   • \(\int_{2}^{7} \frac{1}{x}\,dx = \left[\ln x\right]_{2}^{7}\)
   • \(= \ln 7 - \ln 2\)
   • \(= \ln\frac{7}{2}\)

2. (b) \(\int_1^{e^4} \frac{3}{x}\,dx\).

   • \(\int_1^{e^4} \frac{3}{x}\,dx = \left[3\ln x\right]_1^{e^4}\)
   • \(= 3\ln e^4 - 3\ln 1\)
   • \(= 3\times4 - 0 = 12\)

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

• \(\int\frac{10}{\cos^2(2x)}\,dx=5\tan(2x)\)
• \(\left[5\tan 2x\right]_0^{\pi/12}=5\tan\frac\pi6-5\tan 0\)
• \(\tan\frac\pi6=\frac{1}{\sqrt3}\), so the value is \(\frac{5}{\sqrt3}=\frac{5\sqrt3}{3}\)
• To 3 s.f.: 2.89

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.

• \(\int_{0}^{5} 9e^{-t^2/8}\,dt\): the integrand has no antiderivative we can write, so use the calculator
• From the calculator: \(22.3\)
• 22.3 litres 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.

• Distance = \(\int_{0}^{4} v\,dt\)
• \(\int_{0}^{4} 2t^{\frac{3}{2}}\,dt = \left[\frac{4}{5}t^{\frac{5}{2}}\right]_{0}^{4}\)
• \(= \frac{128}{5} - 0\)
• \(= \frac{128}{5}\)
• Distance = 25.6 m

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 \(t = \ln 4\).

• Added = \(\int_0^{\ln 4} 6e^{t}\,dt\)
• \(= \left[6e^t\right]_0^{\ln 4}\)
• \(= 6e^{\ln 4} - 6e^0 = 6\times4 - 6\times1\)
• \(= 18\): 18 thousand bacteria

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

• \(\int_1^k \frac{2}{x}\,dx = \left[2\ln x\right]_1^k = 2\ln k - 0\)
• \(2\ln k = \ln 25\), so \(\ln k^2 = \ln 25\)
• \(k^2 = 25\), and \(k > 1\), so \(k = 5\)

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.

• \(\int x^{\frac{1}{2}} + \frac{6}{x}\,dx = \frac{2 x^{\frac{3}{2}}}{3} + 6\ln x\)
• \(\left[\frac{2 x^{\frac{3}{2}}}{3} + 6\ln x\right]_{1}^{4} = \left(\frac{16}{3} + 6\ln 4\right) - \left(\frac{2}{3} + 6\ln 1\right)\)
• \(= \frac{14}{3} + 6\ln 4\)
• \(= \frac{14}{3} + 6\ln 4 \approx 13.0\)