Chapter 5

On which derivative rule is the Substitution Rule based?

If \(f\) is an odd function, why is \(\int_{-a}^{d} f(x) d x=0 ?\)

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

What does net area measure?

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 \(0 \leq t \leq 5\).

Why is the Substitution Rule referred to as a change of variables?

If \(f\) is an even function, why is \(\int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x ?\)

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

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

The composite function \(f(g(x))\) consists of an inner function \(g\) and an outer function \(f\). If an integrand includes \(f(g(x)),\) which function is often a likely choice for a new variable \(u ?\)

Is \(x^{12}\) an even or odd function? Is \(\sin x^{2}\) an even or odd function?

Explain in words and write mathematically how the Fundamental Theorem of Calculus is used to evaluate definite integrals.

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?

Suppose you want to approximate the area of the region bounded by the graph of \(f(x)=\cos x\) and the \(x\) -axis between \(x=0\) and \(x=\pi / 2 .\) Explain a possible strategy.

Find a suitable substitution for evaluating \(\int \tan x \sec ^{2} x d x\) and explain your choice.

Explain how to find the average value of a function on an interval \([a, b]\) and why this definition is analogous to the definition of the average of a set of numbers.

Let \(f(x)=c,\) where \(c\) is a positive constant. Explain why an area function of \(f\) is an increasing function.

Suppose that \(f(x) < 0\) on the interval \([a, b] .\) Using Riemann sums, explain why the definite integral \(\int_{a}^{b} f(x) d x\) is negative.

Explain how Riemann sum approximations to the area of a region under a curve change as the number of subintervals increases.

When using a change of variables \(u=g(x)\) to evaluate the definite integral \(\int_{a}^{b} f(g(x)) g^{\prime}(x) d x,\) how are the limits of integration transformed?

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.

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?

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

If the change of variables \(u=x^{2}-4\) is used to evaluate the definite integral \(\int_{2}^{4} f(x) d x,\) what are the new limits of integration?

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 \(R\).

Evaluate \(\int_{0}^{2} 3 x^{2} d x\) and \(\int_{-2}^{2} 3 x^{2} d x\)

Suppose the interval [2,6] is partitioned into \(n=4\) subintervals with grid points \(x_{0}=2, x_{1}=3, x_{2}=4, x_{3}=5,\) and \(x_{4}=6\) Write, but do not evaluate, the left, right, and midpoint Riemann sums for \(f(x)=x^{2}\).

Find \(\int \cos ^{2} x d x\).

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

Explain in words and express mathematically the inverse relationship between differentiation and integration as given by Part 1 of the Fundamental Theorem of Calculus.

Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval \([a, b] ?\) Explain.

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

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

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

Does a left Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and increasing on an interval \([a, b] ?\) Explain.

Find an antiderivative of the following functions by trial and error. Check your answer by differentiating. $$f(x)=(x+1)^{12}$$

Use symmetry to evaluate the following integrals. $$\int_{-2}^{2}\left(3 x^{8}-2\right) d x$$

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.

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 geometry to find a formula for \(\int_{0}^{a} x d x,\) in terms of \(a\)

Find an antiderivative of the following functions by trial and error. Check your answer by differentiating. $$f(x)=e^{3 x+1}$$

Use symmetry to evaluate the following integrals. $$\int_{-\pi / 4}^{\pi / 4} \cos x d x$$

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

If \(f\) is continuous on \([a, b]\) and \(\int_{a}^{b}|f(x)| d x=0,\) what can you conclude about \(f ?\)

Find an antiderivative of the following functions by trial and error. Check your answer by differentiating. $$f(x)=\sqrt{2 x+1}$$

Use symmetry to evaluate the following integrals. $$\int_{-2}^{2}\left(x^{9}-3 x^{5}+2 x^{2}-10\right) d x$$

The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph off and the \(x\) -axis on the interval using a left, right, and midpoint Riemann sum with \(n=4\). $$f(x)=-2 x-1 \text { on }[0,4]$$

Find an antiderivative of the following functions by trial and error. Check your answer by differentiating. $$f(x)=\cos (2 x+5)$$

Use symmetry to evaluate the following integrals. $$\int_{-\pi / 2}^{\pi / 2} 5 \sin x d x$$

The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into \(n\) subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. $$v=e^{t}(\mathrm{m} / \mathrm{s}), \text { for } 0 \leq t \leq 3 ; n=3$$