Definite Integrals Questions and Answers

11. Use the washer method to the find the volume of the solid of revolution formed by revolving the region between y = r² and y = x + 2 around the z-axis. Set the two functions equal to each other to determine the limits of integration. Draw a sketch to determine the upper and lower functions.

4. Determine the area of the region bounded between the two curves y = 1 and y = r²- 3. Draw a sketch of the situation to help determine the upper and lower functions. Set the two functions equal to each other to determine the limits of integration.

Enter a whole or decimal number, or DNE for does not exist of undefined. lim 2-09 I (2+15=9¹)-( 0

Solve sin(x) There are two solutions, A and B, with A<B A = __ 0.6 on 0 < x < 2π B = Give your answers accurate to 3 decimal places

All edges of a cube are expanding at a rate of 4 centimeters per second. (a) How fast is the volume changing when each edge is 1 centimeter(s)? cm³/sec (b) How fast is the volume changing when each edge is 12 centimeters? cm³/sec

Draw the following graph on the interval - 135° < x < 75°: y = cos(-2x)

Use interval notation to express the inequality shown in the graph. -12 -10 -8 OA. [4,00) OB. (4,00) OC. (-∞0,4] OD. (-∞0,4) -6 Select the interval notation below that is represented by the graph. -2 ... 0 2 4 6 8

10. Consider the length of the graph of f(x) = == from (1,5) to (5,1). Approximate the length by finding the sum X of the lengths of 4 line segments, as shown in the following figure. Round your final answer to the nearest tenth of a unit. 5- 4- 3- 2- 1- (1.5) 12 (5,1) The total length of the segment is units.

1 3 2 1 a = To find the blue shaded area above, we would calculate: S. Where: f(x) dx = area f(x) = 2 3 4 Area = b=

Find the volume formed by rotating the region enclosed by: y = x³ + 1, the y-axis, and y = 10 about the y-axis

Write the equation of the ellipse with center at (-3, 5), horizontal minor axis length of 14, and c-6√2. Show at least one line of work and then write the equation in standard form.

If 5 10 ["* f(z)dz = " – [₁ f(a [°¸ ƒ(x)dx + [ªº f(x)dx − ª f(x)dx, 3 5 what are the bounds of integration for the first integral? a= and b =

m [ f(x) dx = 1/1/1 Let x be a continuous random variable over [a,b] with probability density function f. Then the median of the x-values is that number m for which f(x) dx = . Find the a median. f(x) = k e-kx, [0,00)

Evaluate 1 cos(x)ln (1 + x²) + x2.8 1+ el.75a-43.1 S (²² -dx.

(1 point) Find the Laplace transform of F(a) =. f(t)=-5us(t) + Gua(t)-3us(t)

Find the area enclosed by the closed curve obtained by joining the ends of the spiral r = 10,0 ≤ 0 ≤ 5.8 by a straight line segment.

Convert the polar coordinate Enter exact values. X = y = 3, 3πT 4 to Cartesian coordinates.

The functions fand g are defined as follows. g(x)=-2x³-3 f(x)=-5x+2 Find f (5) and g (-3). Simplify your answers as much as possible. f (5) = [] g (-3) = 0 8 X

Express the limit lim(2(7)² - 3(a)¹) Aa, over [2, 4] as an integral. [ f(a)dx. Provide a, b and f(x) in the expression a= b , f(x)

Suppose f''(x) = -54x +36 sin(2x), f'(0) = -9, and f(0) = -1. Then f(x) =

Find the area inside one leaf of the rose: r = 5 sin(40)

Convert the Cartesian coordinate (6,-4) to polar coordinates, 0≤ 0 < 2π, r> 0 Enter exact value. 0

2 This figure shows segments of the graphs of y = 2 √√da O√10dx □f √10dx 10 So √dx 3 These line segments are congruent; their lengths are equal. Which TWO integral expressions can be used to calculate the length of the line segments? 3x and y = 3x and y = x.

what does the following integral equal? -32 96 [6f(x) + 2g(x)-h(x)]dx = -32 √ ( -96 -32 -96 f(x)da = 8 and 9(z)dz = 18 -96 and -32 [h(x)dx=24

If cos (t) = -3, evaluate cos evaluate cos (−t) and sec (−t) 8 cos (-t)

Consider the function f(x) = 5x¹0 + 8x - 3x¹ - 5. An antiderivative of f(x) is F(x) = Ax" + Bxm + CaP + Da² where A is and n is and B is and m is and C is and p is and Dis and q is

1. Evaluate fff8zdV where E is the region bounded by z = 2x² + 2y²-4 and z=5-x² - y² in the E 1st octant.

Use Matrices A, B, C, and D to answer questions 10-15 when possible. When not possible explain using mathematical language. 3 7 2 [-3 -1 4 # # --+] A: B= 4 9 -1 5 D 2 0 -2 12 3 7 2 -4 1 10. What is C22? 11. IDI 12. C-¹

Jan and Monica are both solving a problem that involves an angle of depression of 22.5° from an airplane to a car. Jan draws a triangle that looks like this: 22.50 Monica draws a triangle that looks like this: 22.50 Who is correct and why? Monica is correct because the angle needs to be in the triangle. O Jan is correct because the angle of depression starts from a horizontal line and swings down.

Solve the following I.V.P. using Laplace Transforms: y" - 4y= u₂(t); y(0) = 0, y'(0) = 0.

Find the formula for the area of the inner loop of r = 4 sin θ – 2. You do not need to actually compute the integral.

Set up the limit definition for the area in the first quadrant under f(x) = 3√x on the interval [0, 8]. Translate the limit definition to the structure of the definite integral and compute.

Estimate the area under the function f(x) = (2 + cos(x)) on the interval [0,2π]to 3 decimal places by using a trapezoidal Riemann sum and then again with Simpson's Rule. Compare these two results to the exact area under the curve.

R is the region bounded by the functions f(x) = - x/2 + 4 and g(x) = x/4 - 7 over the interval [a, b] where a = -5 and b= 0. Represent the area A of R by writing an integral with respect to x. You do not need to simplify.

Stacy's credit card has an APR of 18 percent. What is the periodic rate? (A) 18% (B) 1.5% (C) 1.8% (D) 15%

The line y=x+2 and the curve y=x^2 contain bounded region between them. Find the volume of the solid generated by revolving the region about the x-axis. A) 72π/5 B) 64π/5 C) 81π/5 D) 47π/5

Determine the area of the region bounded by the curves x = (y^2) and x + 2y^2 = 3. A) 4 sq units B) 1 sq units C) 2 sq units D) 3 sq units

Find the area of the region bounded by the equations y = x^2 and y=6x - x^2. Note: theses are conics. A) 9 sq units B) 27 sq units C) -18 sq units D) 6 units

The curve y^2 = x, the line x = 2, and the x-axis form the sides of the bounded region. Find the volume of the solid generated by revolving about the x-axis. A) π B) π/2 C) π/4 D) 2π

Find the area bounded by the curves y=6x - x^2 and y= x^2 - 2x. A)128/3 B) 57/3 C) 64/3 D) 101/3

The area bounded by the curve x^2 = 8y, the line y = 0, and the line x = 4 revolves about the y-axis. Find the volume. A) 12π B) 16π C) 18π D) 20π

Suppose v(t) = -t² + 4t - 6 on the interval (1,5). What is the displacement on [1,5]? 1) 12.333 2) 16.733 3) 17.333 4) 56.767

Suppose a function y=f(x) is given with f(x)≥0 for 0 ≤ x ≤ 4. if the area is bounded by the curves y=f(x), y=0, x=0 and x=4 is revolved about the x-axis, then the volume of the resulting solid would best be computed by the method of... A) disks/washers B) shells C) crossections D) polar area

Find the area bounded by the parabola x^2 = 4y and line y = 4. A) 10.67 B) 9.167 C) 18.33 D) 21.33

A certain butterfly population changes at the rate B(t) = 16.2(1.4)ᵗ butterflies per week, where t represents time in weeks. If the butterfly population is 122 at t=0, then how many butterflies are there on week 3? Round to the nearest whole number. 1) 206 2) 187 3) 84 4) 38

(t) (hours) R(t) (gallons per hour) 0 9.6 3 10.4 6 10.8 9 11.2 12 11.4 15 11.3 18 10.7 21 10.2 24 9.6 The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time 1. The table above shows the rate as measured every 3 hours for a 24-hour period. (a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate ∫₀²⁴ R(e)dt. Using correct units, explain the meaning of your answer in terms of water flow, (b) Is there some time 1, 0 < 1< 24, such that R'(t) = 0 ? Justify your answer.

A room is being heated at the rate of r(t) = (0.3t)¹/², where r(t) is measured in degrees Fahrenheit per minute. If the temperature in the room at 6:00 AM is 50 degrees, what is the formula that will describe the temperature at 9:00 AM? Where T(t) = temperature at time t. Do not evaluate the integral.

(a) Sketch the function f(x)=x- 4 from x= -2 to x = 10. (b) Approximate the signed area for f(x) on (-2, 10) by using right hand sums with n = 3. (c) Is your answer in (b) an overestimate or an underestimate of the actual signed area? (d) On the graph that you drew in part (a), sketch the rectangles that you used in part (b).

What is the total change of f(x) if f'(x)=sin(x) from [0, π]?

R is the region bounded by the functions f(x) = 1 - 2Vx and g(x) = -(2x/5) - (3/5). Find the area A of R. Enter an exact answer. Provide your answer below: A = ____ units^2