Problem 20
In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{\sin x}^{\cos x} t^{2} d t$$
Problem 20
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { \cos x } ^ { \pi / 2 } \cos x d x$$
Problem 21
In Exercises \(21-26,\) construct a function of the form \(y=\int^{x} f(t) d t+C\) that satisfies the given conditions. $$\frac{d y}{d x}=\sin ^{3} x,\( and \)y=0\( when \)x=5$$
Problem 21
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 0 } ^ { 1 } e ^ { x } d x$$
Problem 22
Volume of Water in a Reservoir A reservoir shaped like a hemispherical bowl of radius 8 \(\mathrm{m}\) is filled with water to a depth of 4 \(\mathrm{m}\) (a) Find an estimate \(S\) of the water's volume by approximating the water with eight circumscribed solid cylinders. (b) It can be shown that the water's volume is \(V=(320 \pi) / 3 \mathrm{m}^{3}\) . Find the error \(|V-S|\) as a percentage of \(V\) to the nearest percent.
Problem 22
In Exercises \(21-26,\) construct a function of the form \(y=\int^{x} f(t) d t+C\) that satisfies the given conditions. $$\frac{d y}{d x}=e^{x} \tan x,\( and \)y=0\( when \)x=8$$
Problem 23
In Exercises \(23-28,\) use areas to evaluate the integral. $$\int_{0}^{b} x d x, \quad b>0$$
Problem 23
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 1 } ^ { 4 } 2 x d x$$
Problem 23
In Exercises \(23-26\) use a calculator program to find the Simpson's Rule approximations with \(n=50\) and \(n=100 .\) $$\int_{-1}^{1} 2 \sqrt{1-x^{2}} d x$$
Problem 23
In Exercises \(21-26,\) construct a function of the form \(y=\int^{x} f(t) d t+C\) that satisfies the given conditions. $$\frac{d y}{d x}=\ln (\sin x+5),\( and \)y=3\( when \)x=2$$
