91Ó°ÊÓ

Suppose an object moves along a line at \(15 \mathrm{m} / \mathrm{s},\) for \(0 \leq t<2\) and at \(25 \mathrm{m} / \mathrm{s}\), for \(2 \leq t \leq 5,\) where \(t\) is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for \(0 \leq t \leq 5\).

What is the geometric meaning of a definite integral if the integrand changes sign on the interval of integration?

Suppose \(F\) is an antiderivative of \(f\) and \(A\) is an area function of \(f\) What is the relationship between \(F\) and \(A ?\)

Under what conditions does the net area of a region equal the area of a region? When does the net area of a region differ from the area of a region?

The linear function \(f(x)=3-x\) is decreasing on the interval \([0,3] .\) Is the area function for \(f\) (with left endpoint 0 ) increasing or decreasing on the interval [0,3]\(?\) Draw a picture and explain.

Approximating displacement The velocity in \(\mathrm{ft} / \mathrm{s}\) 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 \(v\) 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).

Use symmetry to evaluate the following integrals. $$\int_{-10}^{10} \frac{x}{\sqrt{200-x^{2}}} d x$$

The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of \(f\) and the \(x\) -axis on the interval using a left, right, and midpoint Riemann sum with \(n=4\) c. Use the sketch in part (a) to show which intervals of \([a, b]\) make positive and negative contributions to the net area. $$f(x)=4-2 x \text { on }[0,4]$$

The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of \(f\) and the \(x\) -axis on the interval using a left, right, and midpoint Riemann sum with \(n=4\) c. Use the sketch in part (a) to show which intervals of \([a, b]\) make positive and negative contributions to the net area. $$f(x)=x^{3} \text { on }[-1,2]$$

Use symmetry to evaluate the following integrals. Draw a figure to interpret your result. $$\int_{0}^{2 \pi} \sin x d x$$