🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning ## Chapter 7 ## APPLICATIONS OF INTEGRATION ## Educators   ### Problem 1 Find the area of the shaded region. $$y=5 x-x^{2}$$ Bailey B. Numerade Educator ### Problem 2 Find the area of the shaded region. $$y=\sqrt{x+2}$$ Bailey B. Numerade Educator ### Problem 3 Find the area of the shaded region. $$x=y^{2}-2$$ Bailey B. Numerade Educator ### Problem 4 Find the area of the shaded region. $$x=y^{2}-4 y$$ Bailey B. Numerade Educator ### Problem 5$5-10=$Sketch the region enclosed by the given curves. Decide whether to integrate with respect to$x$or$y .$Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$y=e^{x}, \quad y=x^{2}-1, \quad x=-1, \quad x=1$$ Aidan J. Numerade Educator ### Problem 6 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to$x$or$y .$Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$y=\sin x, y=x, \quad x=\pi / 2, \quad x=\pi$$ Bailey B. Numerade Educator ### Problem 7$5-10=$Sketch the region enclosed by the given curves. Decide whether to integrate with respect to$x$or$y .$Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$y=(x-2)^{2}, \quad y=x$$ Bailey B. Numerade Educator ### Problem 8$5-10=$Sketch the region enclosed by the given curves. Decide whether to integrate with respect to$x$or$y .$Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$y=\sin x, \quad y=2 x / \pi, \quad x \geqslant 0$$ Bailey B. Numerade Educator ### Problem 9$5-10=$Sketch the region enclosed by the given curves. Decide whether to integrate with respect to$x$or$y .$Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$x=1-v^{2} \quad x=v^{2}-1$$ Bailey B. Numerade Educator ### Problem 10$5-10=$Sketch the region enclosed by the given curves. Decide whether to integrate with respect to$x$or$y .$Draw a typical approximating rectangle and label its height and width. Then find the area of the region. $$4 x+y^{2}=12, \quad x=y$$ Bailey B. Numerade Educator ### Problem 11$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=12-x^{2}, \quad y=x^{2}-6$$ Bailey B. Numerade Educator ### Problem 12$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=x^{2}, \quad y=4 x-x^{2}$$ Bailey B. Numerade Educator ### Problem 13$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=e^{x}, \quad y=x e^{x}, \quad x=0$$ Bailey B. Numerade Educator ### Problem 14$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=\cos x, \quad y=2-\cos x, \quad 0 \leqslant x \leqslant 2 \pi$$ Bailey B. Numerade Educator ### Problem 15$11-20=$Sketch the region enclosed by the given curves and find its area. $$x=2 y^{2}, \quad x=4+y^{2}$$ Bailey B. Numerade Educator ### Problem 16$11-20=$Sketch the region enclosed by the given curves and find its area. $$x=y^{4}, \quad y=\sqrt{2-x}, \quad y=0$$ Bailey B. Numerade Educator ### Problem 17$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=\cos \pi x, \quad y=4 x^{2}-1$$ Bailey B. Numerade Educator ### Problem 18$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=|x|, \quad y=x^{2}-2$$ Bailey B. Numerade Educator ### Problem 19$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=1 / x, \quad y=x, \quad y=\frac{1}{4} x, \quad x>0$$ Bailey B. Numerade Educator ### Problem 20$11-20=$Sketch the region enclosed by the given curves and find its area. $$y=\frac{1}{4} x^{2}, \quad y=2 x^{2}, \quad x+y=3, \quad x \geqslant 0$$ Bailey B. Numerade Educator ### Problem 21 Sketch the region that lies between the curves$y=\cos x$and$y=\sin 2 x$and between$x=0$and$x=\pi / 2 .$Notice that the region consists of two separate parts. Find the area of this region. Bailey B. Numerade Educator ### Problem 22 Graph the curves$y=x^{2}-x$and$y=x^{3}-4 x^{2}+3 x$on a common screen and observe that the region between them consists of two parts. Find the area of this region. Bailey B. Numerade Educator ### Problem 23 Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $$y=\frac{2}{1+x^{4}}, \quad y=x^{2}$$ Bailey B. Numerade Educator ### Problem 24$3-26=$Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $$y=e^{1-x^{2}}, \quad y=x^{4}$$ Bailey B. Numerade Educator ### Problem 25$3-26=$Graph the region between the curves and use your calculator to compute the area correct to five decimal places.$y=\tan ^{2} x, \quad y=\sqrt{x}$ Bailey B. Numerade Educator ### Problem 26$3-26=$Graph the region between the curves and use your calculator to compute the area correct to five decimal places. $$y=\cos x, \quad y=x+2 \sin ^{4} x$$ Bailey B. Numerade Educator ### Problem 27 Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use Simpson's Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds. Lucía G. Numerade Educator ### Problem 28 Two cars,$A$and$B,\$ start side by side and accelerate from
rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side. Bailey B.