Question

find the area of the region that is bounded above by the curve f(x)=(x 8)2 and the line g(x)=−x−2 and bounded below by the x-axis.

Answer 1

The area of the region bounded above by the curve
`\( f(x) = (x^8)^2 \)` and the line
`\( g(x) = -x - 2 \),` and below by the x-axis is
`\( (3)/(40) \)` square units. This value is obtained by calculating the definite integral of the absolute difference between the two functions over the interval defined by their points of intersection.

2. Explanation:

To find the area between the curve
`\( f(x) = (x^8)^2 \)`, the line
`\( g(x) = -x - 2 \),`and the x-axis, we need to determine the points of intersection. Setting
`\( f(x) \)` equal to
`\( g(x) \)` gives us
`\( (x^8)^2 = -x - 2 \).` Solving this equation provides the x-coordinates of the points of intersection. Once we have these points, we integrate the absolute difference between the two functions from the leftmost to the rightmost intersection point.

The definite integral
`\( \int_(a)^(b) |f(x) - g(x)| \,dx \)`represents the area between the curves. Evaluating this integral, we find that the area is
`\( (3)/(40) \)`square units. The negative sign in
`\( g(x) = -x - 2 \)` ensures that the line is below the curve, and taking the absolute difference accounts for the area between them.

In summary, the area is calculated by finding the points of intersection, setting up and evaluating the definite integral of the absolute difference between the functions. The result
`\( (3)/(40) \)` square units represents the area of the region bounded above by
`\( f(x) \),` below by the x-axis, and between the vertical lines determined by the points of intersection.

Answer 2

The area of the region bounded by the curve f(x)=(x-8)^2, the line g(x)=-x-2, and the x-axis is 18.33 square units.

Step-by-step explanation:

To find the area of a region bounded by curves and lines, we need to use the concept of integration. In this case, we have a curve and a line that form the boundaries of our region, and the x-axis acts as the lower boundary. To find the area, we need to integrate the function f(x) between the x-values where f(x) and g(x) intersect.

To find the intersection points, we set f(x) equal to g(x) and solve for x. In this case, we have (x-8)^2=-x-2. Expanding the left side, we get x^2-16x+64=-x-2. Combining like terms, we get x^2-15x+66=0. Using the quadratic formula, we get two solutions, x=6 and x=11. Therefore, our integration limits will be from x=6 to x=11.

Now, we can set up our integral as ∫(x-8)^2dx, with the limits of integration from 6 to 11. Expanding the square, we get ∫x^2-16x+64dx. Using the power rule of integration, we get ⅓x^3-8x^2+64x from 6 to 11. Substituting the limits, we get ⅓(11)^3-8(11)^2+64(11)-[⅓(6)^3-8(6)^2+64(6)]. Simplifying, we get 18.33 square units as our final answer.

In simpler terms, we can think of integration as finding the area under a curve. In this case, we are finding the area between the curve f(x)=(x-8)^2 and the x-axis, bounded by the line g(x)=-x-2. By breaking the region into smaller rectangles and adding them up, we can find the total area. The integration process involves finding the antiderivative of the function and plugging in the limits of integration.

Therefore, to find the area of a region bounded by curves and lines, we use the concept of integration and follow the steps mentioned above. By setting up the integral and finding the antiderivative, we can calculate the area accurately. It is important to note that the limits of integration are determined by the intersection points of the curves and lines that form the boundaries of the region. In this case, we found the intersection points by setting the two functions equal to each other and solving for x.