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

Evaluate these definite integrals without a calculator.

1. (a) \(\int_{0}^{32} x^{\frac{1}{5}}\,dx\);

   • \(\int_{0}^{32} x^{\frac{1}{5}}\,dx = \left[\frac{5}{6}x^{\frac{6}{5}}\right]_{0}^{32}\)
   • \(= \frac{160}{3} - 0\)
   • \(= \frac{160}{3}\)

2. (b) \(\int_{1}^{27} x^{- \frac{2}{3}}\,dx\).

   • \(\int_{1}^{27} x^{- \frac{2}{3}}\,dx = \left[3x^{\frac{1}{3}}\right]_{1}^{27}\)
   • \(= 9 - 3\)
   • \(= 6\)

Evaluate these definite integrals without a calculator.

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

   • \(\int_{-4}^{1} x^{3}\,dx = \left[\frac{1}{4}x^{4}\right]_{-4}^{1}\)
   • \(= \frac{1}{4} - 64\)
   • \(= - \frac{255}{4}\)

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

   • \(\int_{-6}^{-3} x^{2}\,dx = \left[\frac{1}{3}x^{3}\right]_{-6}^{-3}\)
   • \(= -9 - \left(-72\right)\)
   • \(= 63\)

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

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

   • \(\int_{\frac{\pi}{4}}^{\frac{3 \pi}{4}} \sin x\,dx = \left[-\cos x\right]_{\frac{\pi}{4}}^{\frac{3 \pi}{4}}\)
   • \(= \left(-\cos \frac{3 \pi}{4}\right) - \left(-\cos \frac{\pi}{4}\right)\)
   • \(= \frac{\sqrt{2}}{2} - \left(- \frac{\sqrt{2}}{2}\right)\)
   • \(= \sqrt{2}\)

2. (b) \(\int_{0}^{\frac{\pi}{3}} \cos x\,dx\).

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

Find the exact value of each integral.

1. (a) \(\int_{0}^{\ln 9} e^{x}\,dx\);

   • \(\int_{0}^{\ln 9} e^{x}\,dx = \left[e^x\right]_{0}^{\ln 9}\)
   • \(= e^{\ln 9} - e^{0}\)
   • \(= 9 - 1\)
   • \(= 8\)

2. (b) \(\int_{\ln 3}^{\ln 7} 5e^{x}\,dx\).

   • \(\int_{\ln 3}^{\ln 7} 5e^{x}\,dx = \left[5e^x\right]_{\ln 3}^{\ln 7}\)
   • \(= 5e^{\ln 7} - 5e^{\ln 3}\)
   • \(= 5\times7 - 5\times3\)
   • \(= 20\)

Find the exact value of each integral.

1. (a) \(\int_{3}^{11} \frac{1}{x}\,dx\);

   • \(\int_{3}^{11} \frac{1}{x}\,dx = \left[\ln x\right]_{3}^{11}\)
   • \(= \ln 11 - \ln 3\)
   • \(= \ln\frac{11}{3}\)

2. (b) \(\int_1^{e^5} \frac{2}{x}\,dx\).

   • \(\int_1^{e^5} \frac{2}{x}\,dx = \left[2\ln x\right]_1^{e^5}\)
   • \(= 2\ln e^5 - 2\ln 1\)
   • \(= 2\times5 - 0 = 10\)

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

• \(\int\frac{9}{\cos^2(3x)}\,dx=3\tan(3x)\)
• \(\left[3\tan 3x\right]_0^{\pi/9}=3\tan\frac\pi3-3\tan 0\)
• \(\tan\frac\pi3=\sqrt3\), so the value is \(3\sqrt3\)
• To 3 s.f.: 5.20

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.

• \(\int_{0}^{4} 14e^{-t^2/6}\,dt\): the integrand has no antiderivative we can write, so use the calculator
• From the calculator: \(29.8\)
• 29.8 grams 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.

• Distance = \(\int_{0}^{8} v\,dt\)
• \(\int_{0}^{8} 3t^{\frac{2}{3}}\,dt = \left[\frac{9}{5}t^{\frac{5}{3}}\right]_{0}^{8}\)
• \(= \frac{288}{5} - 0\)
• \(= \frac{288}{5}\)
• Distance = 57.6 m

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 \(t = \ln 7\).

• Added = \(\int_0^{\ln 7} 4e^{t}\,dt\)
• \(= \left[4e^t\right]_0^{\ln 7}\)
• \(= 4e^{\ln 7} - 4e^0 = 4\times7 - 4\times1\)
• \(= 24\): 24 thousand views

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

• \(\int_1^k \frac{3}{x}\,dx = \left[3\ln x\right]_1^k = 3\ln k - 0\)
• \(3\ln k = \ln 64\), so \(\ln k^3 = \ln 64\)
• \(k^3 = 64\), and \(k > 1\), so \(k = 4\)

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.

• \(\int x^{\frac{1}{3}} + \frac{4}{x}\,dx = \frac{3 x^{\frac{4}{3}}}{4} + 4\ln x\)
• \(\left[\frac{3 x^{\frac{4}{3}}}{4} + 4\ln x\right]_{1}^{8} = \left(12 + 4\ln 8\right) - \left(\frac{3}{4} + 4\ln 1\right)\)
• \(= \frac{45}{4} + 4\ln 8\)
• \(= \frac{45}{4} + 4\ln 8 \approx 19.6\)