Applications of the Intergral

This unit deals with three main ideas:

Applications Involving Motion Along a Line

Area Between the Graphs

Disk and Washer Method: Volume of Revolution

Use an integral to find the total distance traveled and the net displacement of an object moving along a line

Determine the average velocity or average acceleration of an object moving along a line

Connect position, velocity, and acceleration of an object moving along a line

The net displacement of an object whose motion begins at t = 1 and ends at t = b is given by

Total distance traveled by the object from t = a to t = b is given by

If v(t) is greater than or equal to zero between a and b, then the net displacement and the total distance traveled are equal.

Remember, speed and velocity are not the same thing. Speed measures how fast an object is moving regardless of its direction. Velocity measures both the speed and the direction of an object. Because of this, velocity can be positive, negative, or zero.

Suppose the velocity of an object moving along a line is given by a function v = v(t) that is continuous on a closed interval [a, b]. The average velocity of the object from the start of time, a, to the end of time, b, is given by

Suppose the velocity of an object moving along a line is given by a function a = a(t) that is continuous on a closed interval [a, b]. The average acceleration of the object from the start of time, a, to the end of time, b, is given by

Recall from unit 5:

• If the velocity (v) or the first derivative is greater than 0, the position x of the object is increasing with t, and the object is moving to the right.

• If the velocity (v) or the first derivative is less than 0, the position x of the object is decreasing with t, and the object is moving to the left.

• If the velocity (v) or the first derivative is 0, the position x of the object is at rest. This is also where the object changes direction as well.

• Speed is increasing when v(t) and a(t) both have the same sign

• Speed is decreasing when v(t) and a(t) have different signs

To find how fast something is going at time t, you will need to find /v(t)/.

Note: It does not matter about the value, only the sign when determining is something is speeding up or slowing down!

Find the area between the graphs of two functions

Suppose we want to find the area, A, of the region bounded by the graphs of y = f(x), y = g(x), x = a, and x = b. In order to find this area, we would need to first be able to graph the equations to determine which function would be the upper most function. Once we do that, we would integrate those two functions either at the intersections or the two vertical lines that are given.

Assuming f(x) is greater than g(x), then the integral would look like

If at any time, g(x) becomes greater than f(x), you will need to create a new integral showing those boundaries and evaluate normally. So, if you are given the interval [a, b] and at some point on the interval at x = c, g(x) becomes greater than f(x), then you would have

Sometimes it will be necessary to partition the area with the y-axis instead of the x-axis. This will happen when finding the area of the region by partitioning the x-axis is not practical. The graphs that form the left and right borders of the region are not the graphs of functions because they will fail the vertical line test.

In this case, you will either be given x = f(y) and x = g(y) or you might have to solve for them because you are given your boundaries in terms of y instead in terms of x. This means your boundaries will be y = c and y = d. Since we do not have a top and a bottom, we will subtract the right most function from the left most function.

Assuming f(y) is greater than g(y), you then have

Use the disk method to find the volume of a solid formed by revolving a region about the x-axis or the y-axis.

Use the washer method to find the volume of a solid formed by revolving a region about the x-axis or the y-axis.

Suppose a region is bounded by the graph of a function y = f(x) that is continuous and nonnegative on the interval [a, b], the x-axis, and the lines x = a and x = b. Now, suppose this region is revolved about the x-axis. This result is a solid of revolution.

If you have a horizontal line that will be revolved around the x-axis or a vertical line that is revolved around the y-axis, then you will form the geometric solid, a right circular cylinder.

If you have a line that where the slope is greater than zero and the y-intercept is zero and you rotate that line about the x-axis, then you will form the geometric solid, a right circular cone.

These two figures you can use the volume formula from geometry to find the volume of the rotated lines.

In order to use the disk method, a function has to be continuous on the closed interval [a, b], then the volume of the solid of revolution obtained by revolving the region bounded by the graph of f, the x-axis, and the lines x = a and x = b about the x-axis is

In order to use the disk method, a function has to be continuous on the closed interval [c, d], then the volume of the solid of revolution obtained by revolving the region bounded by the graph of f, the y-axis, and the lines y = c and y = d about the y-axis is

If you have more than one function and you are revolving around the x-axis, you will use the washer method. The washer method will have a gap in the middle of the solid. The disk method is be solid throughout.