91Ó°ÊÓ

Find the left endpoint, right endpoint, and midpoint approximations of the area under the curve \(y=e^{x}\) over the interval [0,5] using \(n=5\) subintervals.

Write each expression in sigma notation. but do not evaluate. $$1+2+2^{2}+2^{3}+2^{4}$$

Find the area under the curve \(y=f(x)\) over the stated interval. $$f(x)=x^{-3 / 5} ;[1,4]$$

Evaluate the definite integral two ways: first by a \(u\) -substitution in the definite integral and then by a \(u\) -substitution in the corresponding indefinite integral. $$\int_{1}^{2}(4-3 x)^{8} d x$$

find the derivative and state a corresponding integration formula. $$\frac{d}{d x}[\sin x-x \cos x]$$

Evaluate the integrals by making appropriate substitutions. $$\int x\left(2-x^{2}\right)^{3} d x$$

Write each expression in sigma notation. but do not evaluate. $$2+4+6+8+\dots+20$$

Simplify the expression and state the values of \(x\) for which your simplification is valid. (a) \(e^{-\ln x}\) (b) \(e^{\ln x^{2}}\) (c) \(\ln \left(e^{-x^{2}}\right)\) (d) \(\ln \left(1 / e^{x}\right)\) (e) \(\exp (3 \ln x)\) (f) \(\ln \left(x e^{x}\right)\) (g) \(\ln \left(e^{x-\sqrt[3]{x}}\right)\) (h) \(e^{x-\ln x}\)

Evaluate the definite integral two ways: first by a \(u\) -substitution in the definite integral and then by a \(u\) -substitution in the corresponding indefinite integral. $$\int_{0}^{8} x \sqrt{1+x} d x$$

Find the area under the curve \(y=f(x)\) over the stated interval. $$f(x)=e^{x} ;[1,3]$$