91Ó°ÊÓ

Explain the statement that a continuous function on an interval [a, b] equals its average value at some point on \((a, b)\)

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating. $$\int \sin ^{3} x \cos x d x, u=\sin x$$

Evaluate \(\frac{d}{d x} \int_{a}^{x} f(t) d t\) and \(\frac{d}{d x} \int_{a}^{b} f(t) d t,\) where \(a\) and \(b\) are constants.

Sketch a graph of \(y=2 x\) on [-1,2] and use geometry to find the exact value of \(\int_{-1}^{2} 2 x d x\)

$$\begin{aligned} &\text { Suppose } \int_{1}^{3} f(x) d x=10 \text { and } \int_{1}^{3} g(x) d x=-20 . \text { Evaluate }\\\ &\int_{1}^{3}(2 f(x)-4 g(x)) d x \text { and } \int_{3}^{1}(2 f(x)-4 g(x)) d x \end{aligned}$$

$$\text { Explain why } \int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)$$

Sketch the function \(y=x\) on the interval [0,2] and let \(R\) be the region bounded by \(y=x\) and the \(x\) -axis on \([0,2] .\) Now sketch a rectangle in the first quadrant whose base is [0,2] and whose area equals the area of \(R\)

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating. $$\int(6 x+1) \sqrt{3 x^{2}+x} d x, u=3 x^{2}+x$$

Suppose the interval [1,3] is partitioned into \(n=4\) subintervals. What is the subinterval length \(\Delta x\) ? List the grid points \(x_{0}, x_{1}, x_{2}\) \(x_{3},\) and \(x_{4} .\) Which points are used for the left, right, and midpoint Riemann sums?

$$\text { Use graphs to evaluate } \int_{0}^{2 \pi} \sin x d x \text { and } \int_{0}^{2 \pi} \cos x d x$$