91Ó°ÊÓ

Give a geometrical explanation of why \(\int_{a}^{a} f(x) d x=0\)

Does the right Riemann sum underestimate or overestimate the area of the region under the graph of a positive decreasing function? Explain.

Explain in words and express mathematically the inverse relationship between differentiation and integration as given by the Fundamental Theorem of Calculus.

Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-2}^{2} x^{9} d x$$

Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?

What identity is needed to find \(\int \sin ^{2} x d x ?\)

Does the left Riemann sum underestimate or overestimate the area of the region under the graph of a positive increasing function? Explain.

Symmetry in integrals Use symmetry to evaluate the following integrals. $$\int_{-200}^{200} 2 x^{5} d x$$

The velocity in feet/second of an object moving along a line is given by \(v=3 t^{2}+1\) on the interval \(0 \leq t \leq 4\). a. Divide the interval [0,4] into \(n=4\) subintervals, [0,1] \([1,2],[2,3],\) and \([3,4] .\) On each subinterval, assume the object moves at a constant velocity equal to the value of \(v\) evaluated at the midpoint of the subinterval and use these approximations to estimate the displacement of the object on [0,4] (see part (a) of the figure). b. Repeat part (a) for \(n=8\) subintervals (see part (b) of the figure).

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.