91Ó°ÊÓ

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\ln 8} e^{x} d x$$

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. $$\int 10^{4 t+1} d t$$

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{1 / 2}^{1}\left(t^{-3}-8\right)$$

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand. show the region in question, and interpret your result. $$\int_{0}^{4}(8-2 x) d x$$

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value an the given interval. $$f(x)=8-2 x \text { on }[0,4]$$

Complete the following steps for the given function, interval, and value of \(n.\) a. Sketch the graph of the function on the given interval. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n}\) c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. $$f(x)=\sqrt{x} \text { on }[1,3] ; n=4$$

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand. show the region in question, and interpret your result. $$\int_{-4}^{2}(2 x+4) d x$$

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{4} t(t-2)(t-4) d t$$

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value an the given interval. $$f(x)=e^{x} \text { on }[0,2]$$

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. $$\int\left(\sin ^{5} x+3 \sin ^{3} x-\sin x\right) \cos x \, d x$$