## Integrals Questions and Answers

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If \int_{1}^{6} f(x) dx = - 10 and \int_{3}^{6} f(x) dx = - 8, then what is the value of \int_{1}^{3} f(x) dx ?

Evaluate the integral: integral_{{square root 3} / {2}}^{1 / 2} 6 / {square root {1 - t^2}} dt.

Evaluate the integral: integral_1^2 {V^3 + 3 V^6} / {V^4} dV.

Evaluate the integral: integral_0^{1 / {square root 3}} {t^2 - 1} / {t^4 - 1} dt.

Evaluate the integral: integral_1^9 {x - 1} / {square root x} dx.

Evaluate the integral: integral_1^2 (1 + 2 y)^2 dy.

The graph of y(x) is shown. Evaluate the integral in term of area: \int\limits_0^9 y(x) \text{d} x .

Evaluate the following integral: \int \frac{\sqrt{25 x^2}}{x} \text{d} x

Evaluate the following integral: \int\limits_{ 2}^{ 1} \frac{x}{\sqrt{4x^2 1}} \text{d} x

Suppose \int_2^5 f(x) dx =6, \int ^9_0 f(x) dx=3 and \int_5^9 f(x) dx =1 a. Find each of the following, assuming that f(x) is continuous. i. \int_5^2 f(x) dx ii. \int_9^2 f(x) dx iii. \int_0^2 f(x) dx b. Sketch a possible graph of a function f(x), that s

The graph of y=g(x) is shown. a. Find the value of each of the following: i. \int_2^6 g(x) dx ii. \int_0^6 g(x) dx b. Sketch two possible antiderivatives of y=g(x) over the interval [0.6]

Evaluate \int _0^6(2t-1)^3 dt using the fundamental theorem of calculus. Round to the nearest hundredth if necessary.

If \int _1^8 f(x) dx = 16 and \int _7^8 f(x) dx = 3.7, find \int _1^7 f(x) dx.

Evaluate the integral by interpreting it in terms of areas: \int _{-2}^2 (8-4x) dx

Evaluate the integral \int_0^{1/2} \frac{\arcsin x}{1 - x^2} \,dx

Evaluate each definite integral using area. Be sure to sketch the regions involved. a) \int_{-1}^3 (2 - |x - 1|)dx b) \int_{-2}^3 f(x) dx, with f(x) = 3 x \leq 0 \sqrt{9-x^2} 0< x\leq 3

Use the Integral Test to test the following series for convergence. Solving using other methods is not acceptable. \sum_{n = 3}^{\infty} \frac{1}{n(\ln n)^{2}}

Use the definition of an improper integral to evaluate the following integrals. If an integral converges, evaluate its value. (a) \int_{0}^{1} \frac{1}{x^{0.9}} dx (b) \int_{1}^{\infty} \frac{2x}{x^{2} + 1} dx

Given f(x) = \int_1^x 1+t^2 \text{d}t . Find f(1) .

For the functions u(t) = \left \langle 1, \cos t, t^2 \right \rangle and v(t) = \left \langle t^2, -2t, 1 \right \rangle , a. Compute \frac d{dt}u(t) and \frac d{dt}v(t) . b. Use the Dot Product Rule to compute \frac d{dt}(u(t) \cdot v(t)) . c. Compute t

Evaluate the definite integral ? 8 3 e 1 / x x 2 d x .

Find the most general antiderivative or indefinite integral. ? 7 0 | x 2 ? 4 | d x

Integrate: \int_{-1}^1 x(x^2+1)^{10}dx

Integrate: \int_0^9 (2x^2+x-8)dx

Suppose f(x) is an even function and \int_{-4}^4 f(x)dx=5. Find \int_{-4}^0 f(x)dx.

Suppose that f and g are continuous and that \int_5^9 f(x)dx=-4 and \int_5^9 g(x)dx=7. \\ Find \int_5^9 5f(x)-3g(x)dx

Find G'(x) if G(x)= \int_{\tan x}^{\sqrt x} t^7dt.

Calculate \int_0^8 f(x) dx using subinterval addition property and appropriate area formula from a plane geometry if \\ f(x)= \left\{\begin{matrix} 2x \ \ \text{if} \ \ 0less than x\leq 1 \\ 7, \ \ \text{if} \ \ 1less thanx \leq 5 \\ x+2, \ \ \text{

Sketch the region enclosed by the given curves. Decide whether which is easier to integrate with respect to x or y. Then find the area of the region. \\ y=6x, \ y=5x^2

Consider the area between the graphs x+y=13 and x+7=y^2. This area can be composed in two different ways using integrals. \\ First of all it can be computed as a sum of two integrals \\ \int_a^b f(x)dx+ \int_b^c g(x)dx \\ where a= {Blank}, b={Blank}, c={B

Solve the following integral: \int_0^{\pi/2} \frac d{dx} (\sin \frac x2 \cos \frac x3)\,dx

Which of the following functions corresponds to the graph? a. f(x)=-e^x b. f(x)=-e^x1 c. f(x) = -e^x+1 d. f(x)=e^{-x}+1

If we follow the wording of the fundamental theorem of calculus, we may arrive at the following result: \int^1_{-1} \frac{dx}{x^4}= \left[-\frac{1}{3x^3} \right]^1_{-1}=- \frac{2}{3} Is the above result correct? Explain your reasoning.

Find the area between the graphs of y = x^{3} + 1 and y = 0.5x on the interval \parenthesis 1,2 \parenthesis.

R is the region below the curve y = x and above the x-axis from x = 0 to x = b, where b is a positive constant. S is the region below the curve y = \cos x and above the x-axis from x = 0 to x = b. For what value of b is the area of R equal to the area o

Express the limit as a definite integral over the indicated interval. Do not calculate the value of this integral. \lim_{mesh \to 0} [ \frac{\sqrt{c_1}}{(c_1)^2 + 1} \Delta t_1 + \frac{\sqrt{c_2}}{(c_2)^2 + 1} \Delta t_2 + ? +\frac{\sqrt{c_n}}{(c_n)^2 +

Evaluate the following integral: \int\limits_{ \infty}^{\infty} \frac{e^2}{1+e^{2x}} \text{d} x .

Let f be a function such that its domain is the closed interval [0, \ 6] and its graph is composed of two semicircles as following: 1. First semicircle (top half of the circle) that has a radius of 2 and its center is located at [2, \ 0]

Find the area (polar) of the region bounded by: r = 7 - 2 sin theta.

a) Suppose that a car drives down a city road and its velocity (in m/s) at time (in seconds) is given by: (v) = (2 t^{1/2}) + 4. Determine the car's average velocity between t = 4 seconds and t = 10 seconds. b) Find the average value of f(x) = (6/x^5) + 9

A cola vending company estimates that their cola will be consumed by people at a rate given by C'(t) = 5 + 0.2e^{0.04t} where C'(t) is in billions of gallons per year. How many billions of gallons of cola will be consumed over the next 12 years?

For nonnegative integers n, let A_x : = \int_0^x \sin^nt\,dt a. For n \geq 2, find a formula for A_n(x) in terms of A_{n-2}(x) . b. Evaluate \int_=^\{\pi/4} \sin^{n-2}t (\frac 1n - \cos^2t)\,dt . Your answer should be in terms of n.

Find the volume of the given solid region in the first octant bounded by the plane 8x+2y+4z=8 and the coordinate planes, using triple integrals.

Verify that F(x) is an antiderivative of the integrand f(x) and use Fundamental Theorem to evaluate the definite integral. \int _{1/2}^3 \frac{1}{x}\ dx, \ F(x) = \ln (x)

Verify that F(x) is an antiderivative of the integrand f(x) and use Fundamental Theorem to evaluate the definite integral. \int _1^5 \frac{1}{x} \ dx, \ F(x) = \ln (x)

Verify that F(x) is an antiderivative of the integrand f(x) and use Fundamental Theorem to evaluate the definite integral. \int _0^3 (x^2+4x-3) \ dx, \ F(x) = \frac{1}{3} x^3+2x^2-3x

Verify that F(x) is an antiderivative of the integrand f(x) and use Fundamental Theorem to evaluate the definite integral. \int _1^3 x^2 \ dx, \ F(x) = \frac{1}{3} x^3

Evaluate A'(x) at x=1,2 and 3. A(x)= A(x) = \int^x_1 2t \ dt

Evaluate A'(x) at x = 1, 2 and 3. A(x) = \int^x_0 2t \ dt

If f(x) is continuous and \int_0^1 f(x) \, dx = 3, then find 3\int_0^{\pi/4} \sec^2 t \cdot f(\tan t) dt .

Evaluate the definite interval. \int_1^4 (5t - 3)dt

Use the definite integral to determine the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis anywhere in the given interval and split the required accordingly. f(x) = 9 - x^2, [0,6]

Evaluate the definite integral. \int_{-2}^{-1} \frac{4x^3 - 3x^2 + 1}{x^2}dx =

Evaluate the definite integral. \int_1^e \frac{(3 \ln x)^2}{x} dx =

Evaluate the definite integral. \int_{-2}^0 e^{2x + 4} dx =

Evaluate the definite integral. \int_0^1 2x(x^2 + 9)^2 dx =

Evaluate the definite integral. \int_{-1}^1 (x^2 + 1)^2 dx =

Find the area of the surface generated by revolving about the x-axis the curve f(x) = 2 \sqrt{1 - x} on [-1,0].

The marginal revenue from box-office sales (in billions of dollars per billion of tickets sold) can be approximated by the function MR(x) = 0.56x + 3.1 where x is the number of tickets sold (in billions). Over the previous decade, about 13 billion tickets

Given the parametric curve C defined by x = \frac{1}{1 + t}, y = \ln(1 + t), set up the integral needed to find the arclength of C from (1,0) to (0.25,\ln 4).