91Ó°ÊÓ

Assume that \(f\) is continuous on \([a, b]\) and $$ \int_{a}^{b} f(x) d x=0 $$ Answer questions \(7-15,\) giving supporting reasons. Does it necessarily follow that \(f(x)=0\) for at least some \(x \in[a, b] ?\)

Find the area between the graph of \(f\) and the \(x\) -axis. $$f(x)=\cos x, \quad x \in\left[\frac{1}{6} \pi, \frac{1}{3} \pi\right]$$

Calculate. $$\int\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right) d x$$

Determine the average value of the function on the indicated interval and find an interior point of this interval at which the function takes on its average value. $$f(x)=3-2 x, \quad x \in[0,3]$$

Calculate the following for each \(F\) given below: \(\begin{array}{llll}\text { (a) } F^{\prime}(-1) & \text { (b) } F^{\prime}(0)\end{array}\) (c) \(F^{\prime}\left(\frac{1}{2}\right)\) (d) \(F^{\prime \prime}(x)\) $$F(x)=\int_{x}^{5} \sqrt{t^{2}+1} d t$$

Find the area between the graph of \(f\) and the \(x\) -axis. $$f(x)=\sin x, \quad x \in\left[\frac{1}{3} \pi, \frac{1}{2} \pi\right]$$

Calculate the following for each \(F\) given below: \(\begin{array}{llll}\text { (a) } F^{\prime}(-1) & \text { (b) } F^{\prime}(0)\end{array}\) (c) \(F^{\prime}\left(\frac{1}{2}\right)\) (d) \(F^{\prime \prime}(x)\) $$F(x)=\int_{x}^{1} t \sqrt{t^{2}+1} d t$$

Evaluate the integral. $$\int_{-2}^{0}(x+1)(x-2) d x$$

Calculate \(L_{f}(P)\) and \(U_{f}(P)\). $$f(x)=\sin x, \quad x \in[0, \pi] ; \quad P=\left\\{0, \frac{1}{6} \pi, \frac{1}{2} \pi, \pi\right\\}$$.

Calculate. $$\int x^{2}\left(1+x^{3}\right)^{1 / 4} d x$$