Application of Integrals

In applications of integrals, definite integration is mainly used because it gives

1. The derivative of a function
2. The equation of the tangent
3. The exact accumulated value over an interval
4. The inverse of a function

Which of the following is the correct procedure for finding the area between two curves?

1. Differentiate both curves and subtract
2. Integrate the product of the curves
3. Integrate (Upper Curve minus Lower Curve) over the interval
4. Integrate (Lower Curve minus Upper Curve) without checking

If two curves intersect at only one point, then the area enclosed between them is

1. Always positive
2. Zero
3. Generally cannot form a closed bounded region
4. Always equal to 1

Find the area enclosed between the curves y = x squared and y = 4 from x = -2 to x = 2

1. 16/3 square units
2. 32/3 square units
3. 8 square units
4. 64/3 square units

The area enclosed between the line y = x and the x-axis from x = 0 to x = 3 is

1. 3 square units
2. 9/2 square units
3. 6 square units
4. 9 square units

The area enclosed by a curve is always

1. Negative
2. Positive or zero
3. Equal to the definite integral without considering sign
4. Always greater than 1

If a curve is symmetric about the y-axis, then the area bounded by the curve from x = -a to x = a is

1. integral from -a to a of f(x) dx
2. 2 x integral from 0 to a of f(x) dx
3. integral from 0 to a of f(x) dx
4. Zero

If two curves intersect at x = a and x = b, then the limits of integration for finding the enclosed area are

1. 0 and 1
2. The points of intersection
3. The maximum values of y
4. Any convenient values

The area enclosed between the curve y = 4 and the x-axis from x = 1 to x = 5 is

1. 4 square units
2. 8 square units
3. 16 square units
4. 20 square units

Find the area enclosed between the curves y = x and y = x squared from x = 0 to x = 1

1. 1/3 square units
2. 1/6 square units
3. 1/2 square units
4. 2/3 square units

The area between the curves y = f(x) and y = g(x), where f(x) >= g(x) on [a, b], is

1. integral from a to b of [f(x) + g(x)] dx
2. integral from a to b of [f(x) - g(x)] dx
3. integral from a to b of f(x)g(x) dx
4. integral from a to b of [g(x) - f(x)] dx

If the graph of a function lies completely below the x-axis between x = a and x = b, then the definite integral from a to b of f(x) dx is

1. Positive
2. Negative
3. Zero
4. Always equal to 1

The area enclosed between the curve y = x squared and the x-axis from x = 0 to x = 2 is

1. 4/3 square units
2. 8/3 square units
3. 4 square units
4. 8 square units

The area bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b is given by

1. f(b) - f(a)
2. integral from a to b of f(x) dx
3. f'(b) - f'(a)
4. integral from a to b of f'(x) dx

The primary application of definite integrals in this chapter is to calculate

1. Derivatives of functions
2. Area bounded by curves
3. Determinants of matrices
4. Limits of functions