33 Lecture

A cylindrical tank is filled with water to a height of 10 meters. The radius of the tank is 5 meters. What is the approximate volume of the water in the tank?

a. 785.4 m^3

b. 1570.8 m^3

c. 1963.5 m^3

d. 3141.6 m^3

Answer: b. 1570.8 m^3

What is the average value of the function f(x) = 3x^2 + 2x + 1 on the interval [0,1]?

a. 2

b. 3

c. 4

d. 5

Answer: c. 4

The region bounded by y = x^2 and y = x is rotated around the y-axis. What is the volume of the resulting solid?

a. 1/6?

b. 1/4?

c. 1/2?

d. 3/4?

Answer: b. 1/4?

A rectangular tank with a length of 4 meters and a width of 2 meters is being filled with water at a rate of 2 cubic meters per minute. How fast is the water level rising when the depth of the water is 3 meters?

a. 1/6 m/min

b. 1/3 m/min

c. 2/3 m/min

d. 1 m/min

Answer: c. 2/3 m/min

The region bounded by y = sin x, y = 0, x = 0, and x = ? is rotated around the x-axis. What is the volume of the resulting solid?

a. 2?

b. 2?/3

c. 4?/3

d. 8?/3

Answer: b. 2?/3

A wire of length 10 meters is bent into the shape of a rectangle. What is the maximum area of the rectangle?

a. 5 m^2

b. 10 m^2

c. 12.5 m^2

d. 25 m^2

Answer: c. 12.5 m^2

The region bounded by y = x^3, y = 0, x = 1, and x = 2 is rotated around the x-axis. What is the volume of the resulting solid?

a. 7/3?

b. 8/3?

c. 9/2?

d. 10/3?

Answer: b. 8/3?

A rectangular tank with a length of 6 meters and a width of 4 meters is being filled with water at a rate of 3 cubic meters per minute. How fast is the water level rising when the depth of the water is 2 meters?

a. 1/3 m/min

b. 1/2 m/min

c. 2/3 m/min

d. 1 m/min

Answer: d. 1 m/min

The region bounded by y = x^2, y = 2x, and x = 2 is rotated around the y-axis. What is the volume of the resulting solid?

a. 8?/15

b. 4?/3

c. 8?/3

d. 16?/15

Answer: a. 8?/15

A particle moves along a straight line such that its position at time

What is the application of definite integral in finding the area under a curve?

Answer: The application of definite integral in finding the area under a curve is that it can be used to calculate the total area enclosed by a function and the x-axis over a specific interval.

How can definite integral be used in finding the average value of a function?

Answer: Definite integral can be used in finding the average value of a function by dividing the integral of the function over a given interval by the length of the interval.

Explain the use of definite integral in calculating work done by a variable force.

Answer: The use of definite integral in calculating work done by a variable force is that it can be used to determine the work done by a force that varies in magnitude and direction over a given distance.

What is the application of definite integral in calculating the center of mass of an object?

Answer: The application of definite integral in calculating the center of mass of an object is that it can be used to determine the coordinates of the point at which the object balances or the point at which the object's mass is evenly distributed.

Explain the use of definite integral in calculating the volume of a solid of revolution.

Answer: The use of definite integral in calculating the volume of a solid of revolution is that it can be used to sum the volume of an infinite number of infinitesimal slices of the solid generated by rotating a function around a given axis.

How can a definite integral be used in finding the distance traveled by an object with variable velocity?

Answer: Definite integral can be used in finding the distance traveled by an object with variable velocity by integrating the velocity function over a given time interval.

Explain the use of definite integral in calculating the probability density function.

Answer: The use of definite integral in calculating the probability density function is that it can be used to determine the probability of a random variable falling within a certain range of values.

What is the application of definite integral in calculating the heat transfer in a system?

Answer: The application of definite integral in calculating the heat transfer in a system is that it can be used to sum the infinitesimal amounts of heat transferred in a system over a given time interval.

Explain the use of definite integral in finding the total charge of a system.

Answer: The use of definite integral in finding the total charge of a system is that it can be used to sum the infinitesimal charges of the system over a given time interval.

What is the application of definite integral in calculating the moment of inertia of an object?

Answer: The application of definite integral in calculating the moment of inertia of an object is that it can be used to sum the infinitesimal contributions of each point in the object to the overall moment of inertia.

Application of Definite Integral

Calculus is a branch of mathematics that deals with the study of continuous change. One of the fundamental concepts in calculus is the definite integral, which is used to calculate the area under a curve. However, the application of definite integrals extends beyond just finding areas. In this article, we will discuss some of the common applications of definite integrals.

Finding the area between two curves:

The definite integral can be used to find the area between two curves. To find the area between two curves, we first find the points of intersection between the two curves. We then set up the integral with the limits of integration as the x-coordinates of the points of intersection. The integrand is then the difference between the two functions.

Calculation of volume:

Definite integrals can be used to calculate the volume of a solid of revolution. A solid of revolution is obtained by rotating a curve around an axis. To calculate the volume of a solid of revolution, we first divide the curve into small segments of equal width. We then approximate the volume of each segment by a cylinder whose volume is given by the formula ?r^2h. We sum up the volumes of all the cylinders and take the limit as the segment width approaches zero. This gives us the exact volume of the solid of revolution.

Calculation of arc length:

The definite integral can be used to calculate the length of a curve. To find the length of a curve, we first divide the curve into small segments of equal width. We then approximate the length of each segment by the length of a straight line segment connecting the two endpoints of the segment. We sum up the lengths of all the straight-line segments and take the limit as the segment width approaches zero. This gives us the exact length of the curve.

Calculation of work:

Definite integrals can be used to calculate the work done by a force acting on an object. To calculate the work done by a force, we first find the displacement of the object. We then calculate the dot product of the force and the displacement vectors. We integrate the dot product over the path of the object to get the total work done by the force.

Calculation of center of mass:

The definite integral can be used to find the center of mass of a two-dimensional object. To find the center of mass, we first divide the object into small segments of equal area. We then approximate the location of the center of mass of each segment by the centroid of the segment. We sum up the products of the mass of each segment and the coordinates of its centroid and then divide by the total mass of the object. This gives us the coordinates of the center of mass of the object.

Calculation of probability:

Definite integrals can be used to calculate probabilities in probability theory. For example, the probability that a random variable takes a value between two given values is given by the definite integral of the probability density function between those two values.

Calculation of fluid flow:

Definite integrals can be used to calculate the flow of fluid through a pipe. To calculate the flow of fluid, we first divide the cross-section of the pipe into small segments of equal width. We then approximate the flow of each segment by a cylinder whose volume is given by the area of the segment times the speed of the fluid. We sum up the volumes of all the cylinders and take the limit as the segment width approaches zero. This gives us the exact flow of fluid through the pipe.

Calculation of heat transfer:

Definite integrals can be used to calculate the heat transfer between two bodies. To calculate the heat transfer, we first find the temperature difference between the two bodies. We then integrate the product of the temperature difference, the area of contact