Definite Integrals Questions and Answers

Evaluate the double integral by first identifying it as the volume of a solid. ∫∫D (3x +2y+6)dA where D = {(x, y)| -1 ≤ x ≤ 1,-2 ≤ y ≤ 2}.

Given that f(x, y) = -x + 1 and D is the triangular region with vertices (0, 0), (0, 2), (6, 2), set up two integrals which evaluate the volume between the region D and the surface f(x, y) in two different ways. Do not evaluate these two integrals.

Evaluate the first partial derivatives of the function at the given point. f(x, y) = 8ex In(y); (0, e) f(0, e) = f(0, e) =

The angle A is found in quadrant IV, such that sin A = -4 7 Determine the exact value of sec A (Decimal answers using a calculator will not be accepted).

Find F' (x) if F(x) = ₀∫x²+1 teᵗ dt. Evaluate 1∫∞ 1/x³ dx. Show all working.

Simplify: (-13-5i) - (-18+2i) 5-7i -31 - 3i -5-3i 234-10i

Use the disk method to find the volume of the solid of revolution formed by revolving the region underneath y = 2x² around the x-axis from z = 0 to z = 4. Draw a sketch.

Set up (but do not compute) the integrals that would represent the area of the region bounded between the two curves y = x³ and y = x² + x. Set the two functions equal to each other to determine the limits of integration. Use a graphing software or calculator to see a picture of the situation. Note that the upper and lower function will change along the interval! Note: I am just having you set up and not compute the integrals because you should notice that the limits of integration are not particularly nice for this question!

If A = 2yzi - (x + 3y − 2)j + (x² + z)k, evaluate ∫∫(▽ x A). n dS, over surface of intersection of the cylinders x² + y² = a², x² + z² = a², which is included in the first octant.

Please estimate the area under the curve c(x) = 1 - x³ for 0 ≤ x ≤ 1 using 4 rectangles with width 1/4. Please show a sketch of the curve, the rectangles, and calculations. (A calculator is allowed.)

Determine a particular solution for the differential equation dy/dt + 2y = 2/π ₀∫∞ [sin(3ω) - 3ω cos(3ω) / ω²] sin(ωt) dω.

Write a formula for the function g that results when the graph of the function f(x)=-7/x is reflected about the y-axis, horizontally compressed by a factor of 3, shifted 1 units left, vertically stretched by a factor of 5, and shifted 4 units up. g(x) =

Compute ∫c F .dr where F(x, y) = (-y³, x³) and C is the counterclockwise oriented unit circle.

Q.18 If the Laplace transform of y(t) is given by Y(s) = L(y(t)) = 5/2(s-1)-2/(s-2)+1/2(s-3) then y(0) + y'(0) =

2 For the given probability density function, over the stated interval, find the requested value. f(x) = x, over the interval [1,3]. Find E(x). A. 13/6 B. 25/12 C. 9/4 D. 5/8

Determine whether or not the function is a probability density function over the given interval. f(x) = 1/5x, [1,4] Yes No

Determine whether the integral is convergent or divergent. ∫e^-5x dx (4 to ∞) Divergent Convergent

Use a definite integral to find an expression that represents the area of the region between the given curve and the x-axis on the interval [0, b]. 15) y = 6x Compute the definite integral as the limit of Riemann sums.

A curve, described by x² + y² + 6y = 0, has a point A at (-3, -3) on the curve. Part A: What are the polar coordinates of A? Give an exact answer. Part B: What is the polar form of the equation? What type of polar curve is this? Part C: What is the directed distance when θ: 4π/3? Give an exact answer.

Let ∫ h(x)dx = 252 ∫ h(x) dx = − 144 Use the given definite integrals above to evaluate each definite integral below. ∫ - 2h(x) dx = ∫ - 3h(x) dx ∫ h(x)/9 dx ∫ h(x) dx =

Consider ∫g(x)dx (1to-9) - ∫g(x)dx(-8to-9) Use properties of definite integrals to rewrite the expression above as a single definite integral.

Let g(x) = ∫2dt (x to-2) Use the limit definition of the definite integral to evaluate the following expressions. g(-2) = g(-4)= g(5) =

Let g(z) = ∫( − 7t + 6)dt Use the limit definition of the definite integral to evaluate the following expressions. g(-5)= g(2)= g(-7)=

Use cylindrical coordinates. Find the mass and center of mass of the S solid bounded by the paraboloid z= 10x^2 + 10y2 and the plane z = a (a > 0) if S has constant density K. m=

Find E(x), E(x²), the mean, the variance, and the standard deviation of the random variable whose probability density function is given below. f(x) = 1/72x, [0,12]

The endpoints of a diameter of a circle are (-9, 6) and (-15, -2). (a) Write an equation of the circle in standard form. (b) Graph the circle. Part 1 of 2 (a) An equation of the circle in standard form is Part: 1 / 2 Part 2 of 2 (b) Graph the circle.

Find the area of the region bounded by the graph of the equation. 25.) y = 1 + 3x² x = -2, x = 1, y = 0

Expand the logarithm. log7(49x³-√z) 2+3 log7 x - 2 log7 z 2+3 log7 x + 1 2log7 z 2+3 log x + 1 2log z log 49 + 3 log x + 1 2log z

Consider the curve C parameterized by r(t) = (2t, -t2) for 0 ≤ t ≤ 1. Compute using the definition of the line integral ∫cF.dr.

Find the area of the surface. the part of the plane 2x + 13y + z = 26 that lies in the first octant

(Coding) In this problem we will find the exponential of best fit (y = c1ec2t) through the points (1, 2), (2, 2), (3, 7), (4, 10), (5. 17). (a) Find the vector of equations r and the Jacobian Dr. (b) Determine (with justification) some initial guess x0

Solve the equation and enter the solution set below. Enter the solutions in order from smallest to largest separated by a comma. Do not enter blank spaces in your answer. x(x - 9) = 6²2

The following number is equivalent to 3 for what value of n? 1/27

When the expression is completely simplified, what is the exponent of 10? (10¹/2) -8x+6

Solve the equation. Enter the solution below.

Simplify the expression. What exponent of 3 results? Do not enter any blank spaces i in your answer. (3-4)3-x

Find the standard equation for the positions of a body moving with a constant acceleration a along a coordinate The following properties are known: i. d²s dt² = a, ii. ds dt = v0 when t = 0, and iii. s = s0 when t = 0, where t is time, s0 is the initial position, and v0 is the initial velocity.

The base of a solid is the region bounded by y=x' and y=4. Find the volume of the solid given that the cross sections perpendicular to the x-axis are: (a) squares, (b) semicircles, (c) equilateral triangles.

Consider the following function. f(x) = cos (8x 9) Find the derivative of the function. Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. The given function is f(x) = cos(8x 9) Differentiate f(x) with respect to x. The function f(x) is strictly monotonic, if f '(x) is f '(x) > 0 or f '(x) < 0✔ f'(x) > 0 or f '(x) < 0 on the domain of f(x). Also to be strictly monotonic, the function f(x) should not take the same value for different values of x. That is, f(x₁) = f(x₂) if X₁ ≠ X2

Find the volume of the solid E described by the solution of this system of inequalities: x² + y² ≤ 1 x² +2² ≤1 y² +z²≤1

Find the moments of inertia I, I, Io for a lamina in the shape of an isosceles right triangle with equal sides of length a if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. (Assume that the coefficient of proportionality is k, and that the lamina lies in the region bounded by x = 0, y = 0, and y = a - x). Ix = Iy= Io=

Find E(x), E(x²), the mean, the standard deviation and variance, over the given interval, of the random variable whose probability density function is given below. f(x) = 1 4, [3,7]

Use Newton's method to approximate the indicated solution of the equation correct to six decimal places. the positive solution of e³x = x + 8

Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

Use a double integral to find the area of the region. one loop of the rose r = 5 cos(3θ)

Find the derivative or indefinite integral as indicated. d/dt ∫ t9 - lnt 7t+4 dt

Find the antiderivative of the given derivative. ds dt = 11t(4t² - 1)³

Find the volume of the given solid. under the plane 9x + 2y - z = 0 and above the region enclosed by the parabolas y = x² and x = y²

Let f (x) = − 4x + 1. Find the inverse of f (x). f¹(x)=-x+1/4 f¹(x) = -4x – 1 Does not exist f¹(x) = -4x+1

Find the volume of the solid enclosed by the paraboloid z = 2 + x² + (y - 2)2 and the planes z = 1, x = -3, x = 3, y = 0, and y = 2.