Question

Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3.

A smooth curve and three shaded regions are graphed on the x y coordinate plane. The curve enters the viewing window in the second quadrant, goes down and right, and passes through the point (−4, 0). The curve continues down and right, reaches a local minimum, then goes up and right to pass through the point (−2, 0). The curve continues up and right, reaches a local maximum, then goes down and right to pass through the point (0, 0). The curve continues down and right, reaches a local minimum, then goes up and right to pass through the point (2, 0). The curve continues up and right and ends in the first quadrant. The first region labeled A is above the curve and below the x-axis between the values of x = −4 and x = −2. The second region labeled B is below the curve and above the x-axis between the x values of x = −2 and x = 0. The third region labeled C is above the curve and below the x-axis between the x values of x = 0 and x = 2.
Find the value of

Answer 1

The value
`\(\int_(-4)^(2) |f(x)| \,dx\)` is 12.

Step-by-step explanation:

To find the value of
`\(\int_(-4)^(2) |f(x)| \,dx\)`, we need to consider the absolute value of the function f(x) since the regions A, B, and C are defined based on the position of the curve for the x-axis.

The integral represents the signed area between the curve and the x-axis over the interval [-4, 2]. The absolute value ensures that the areas above and below the x-axis contribute positively to the integral. We can break the interval into three parts corresponding to regions A, B, and C. The absolute values account for the areas above the curve in A and C, as well as the area below the curve in B.

`\[\int_(-4)^(2) |f(x)| \,dx = \int_(-4)^(-2) f(x) \,dx + \int_(-2)^(0) -f(x) \,dx + \int_(0)^(2) f(x) \,dx\]`

Given that the areas of A, B, and C are each 3, the sum of these absolute values is 3 + 3 + 3 = 9. Therefore, the value of
`\(\int_(-4)^(2) |f(x)| \,dx\)` is 12, considering the signed areas of the regions under the curve.

Complete Question: Each of the regions A, B, and C is bounded by the graph of f, and the x-axis has area 3.

A smooth curve and three shaded regions are graphed on the x-y coordinate plane. The curve enters the viewing window in the second quadrant, goes down and right, and passes through the point (−4, 0). The curve continues down and right, reaches a local minimum, then goes up and right to pass through the point (−2, 0). The curve continues up and right, reaches a local maximum, then goes down and right to pass through the point (0, 0). The curve continues down and right, reaches a local minimum, then goes up and right to pass through the point (2, 0). The curve continues up and right and ends in the first quadrant. The first region labeled A is above the curve and below the x-axis between the values of x = −4 and x = −2. The second region labeled B is below the curve and above the x-axis between the x values of x = −2 and x = 0. The third region labeled C is above the curve and below the x-axis between the x values of x = 0 and x = 2.

Find the value
`\(\int_(-4)^(2) |f(x)| \,dx\)`?

Answer 2

The question addresses calculating areas between a function graph and the x-axis using definite integrals, with each region A, B, and C having an area of 3. The sign of the integral indicates whether the graph is above or below the x-axis.

Step-by-step explanation:

The student seems to be asking about properties of a function graph and how it relates to calculating areas between the graph and the x-axis.

To find the area under the graph of a function, we often use the definite integral of the function over a given interval.

Since each of the regions A, B, and C has an area of 3 units, the integral of the function over each respective interval would also be equal to 3.

The curve described appears to pass through the x-axis at points (−4, 0), (−2, 0), (0, 0), and (2, 0), creating three regions A, B, and C where the areas under the curve are either above or below the x-axis.

For regions A and C, where the areas are above the curve and below the x-axis, the integral of the function would take on a negative value, meaning we would be looking at the negative areas.

In contrast, for region B, where the area is below the curve and above the x-axis, the integral would be positive.

When calculating the definite integral to find the area, it is important to consider the sign, as this indicates whether the area is below or above the x-axis.