91Ó°ÊÓ

\(15-18=\) Find the average value of the function on the given interval. $$g(x)=\cos x, \quad[0, \pi / 2]$$

\(15-18=\) Find the average value of the function on the given interval. $$f(\theta)=\sec \theta \tan \theta, \quad[0, \pi / 4]$$

\(15-18=\) Express the limit as a definite integral on the given interval. $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{x_{i}^{*}}{\left(x_{i}^{*}\right)^{2}+4} \Delta x,[1,3]$$

Evaluate the indefinite integral. $$\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$$

\(1-30\) " Evaluate the integral. $$\int_{0}^{\pi / 3} \frac{\sin \theta+\sin \theta \tan ^{2} \theta}{\sec ^{2} \theta} d \theta$$

Let \(A\) be the area under the graph of an increasing continuous function \(f\) from \(a\) to \(b,\) and let \(L_{n}\) and \(R_{n}\) be the approximations to \(A\) with \(n\) sub intervals using left and right endpoints, respectively. (a) How are \(A, L_{n},\) and \(R_{n}\) related? (b) Show that $$R_{n}-L_{n}=\frac{b-a}{n}[f(b)-f(a)]$$ Then draw a diagram to illustrate this equation by showing that the \(n\) rectangles representing \(R_{n}-L_{n}\) can be reassembled to form a single rectangle whose area is the right side of the equation. (c) Deduce that $$R_{n}-A<\frac{b-a}{n}[f(b)-f(a)]$$

\(19-23\) . Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. $$\int_{2}^{5}(4-2 x) d x$$

Evaluate the indefinite integral. $$\int e^{x} \sqrt{1+e^{x}} d x$$

\(1-30=\) Evaluate the integral. $$\int_{0}^{1} \cosh t d t$$

\(19-20=\) (a) Find the average value of \(f\) on the given interval (b) Find \(c\) such that \(f_{\text { fore }}=f(c) .$$19-20=\) (a) Find the average value of \(f\) on the given interval (b) Find \(c\) such that \(f_{\text { fore }}=f(c) .\) (c) Sketch the graph of \(f\) and a rectangle whose area is the same as the area under the graph of \(f .\) $$f(x)=(x-3)^{2}, \quad[2,5]$$