Definite Integrals Questions and Answers

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8. 64y=x^3, y=0, x=8

Find the volume generated by rotating the area bounded by the graph of the following set of equations around the y-axis. y = 3x² +2, x = 1, x = 3 The volume of the solid is

Find the volume generated by rotating the area bounded by the graph of the following set of equations around the y-axis. y = 3x² +2₁ x= 1, x = 3

Find the volume generated by rotating the area bounded by the graph of the following set of equations around the y-axis.

A particle is moving with acceleration given by a(t) = 6t-12 and an initial velocity of 9 m/s. (a) Find the velocity of the particle at time t. (b) Find the velocity of the particle after 3 seconds. (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total displacement of the particle during the first 8 seconds. (f) Find the total distance travelled by the particle during the first 8 seconds.

Find the volume generated by rotating the area bounded by the graph of the following set of equations around the y-axis. y = 3x² + 2, x= 2₁ x = 3 The volume of the solid is cubic units.

The infectivity of a certain pathogen is quantified by measuring the area between two curves. The greater the area between the curves, the greater the infectivity of the disease and this quantification provides a means to compare infectivity among various different pathogens. As an example, consider the function f(t) = -0.5t(t + 1)(t - 21) which represents number of infected cells carrying a particular pathogen in the blood of a patient as a function of time t, where t is measured in days. Suppose a patient is considered contagious during the period from time t = 3 days to t = 18 days. Compute the infectivity of this particular disease during the contagious period by finding the area between the function f(t) and the line segment connecting the points (3,f(3)) and (18,f(18)). Include a graph of the area computed.

Refer to the figure and find the volume generated by rotating the given region about the specified line.

The graph of the function f on the interval 0 s x s9 consists of three line segments and the point (4,-4), as shown in the figure. It is given that Sog(x) dx = Find the value of Part B: Find the value of Part C: Find the value of f5₂f(x) dx or explain why the integral does not exist. (3 points) g(x) dx. Show the work that leads to your answer.

A graph consists of the line y=5 for 1 sxs 5 and y=-1 for-3 sxs 1. Find the accumulation of change under the curve from x=-3 to x = 5 using geometric formule -4 square units 4 square units 16 square units 24 square units

Let x)=x², and compute the Riemann sum off over the interval [4, 51, choosing the representative points to be the midpoints of the subintervals and using the following number of subintervals (n). (Round your answers to two decimal places.) (a) two subintervals of equal length (n=2) (b) five subintervals of equal length (n = 5) (c) ten subintervals of equal length (n = 10) (d) Can you guess at the area of the region under the graph of fon the interval [4, 5]?

How long will it take the bar to reach 99° C? (Round your answer to one decimal place.) sec A small metal bar, whose initial temperature was 30° C, is dropped into a large container of boiling water. How long will it take the bar to reach 80° C if it is known that its temperature increases 2° during the first second? (The boiling temperature for water is 100° C. Round your answer to one decimal place.) sec

The curve y² = 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/2 B π/4 C 2π D π

The volume produced by revolving about the x-axis the region above the curve y= x³, below the line y = 1 and between x = 0 and x = 1 is ... A π/42 B π/7 C √2/2π D 6π/7

Use a double integral in polar coordinates to find the volume V of the solid bounded by the graphs of the equations. z =√x² + y² z = 0 x² + y²= 4

Set up a triple integral for the volume of the solid. Do not evaluate the integral. The solid in the first octant bounded by the coordinate planes and the plane z = 5 - x - y