How do I find the area of the region bounded by the graphs of the given equations?

y=x+20; y=x^2

The graphs intersect at (-4,20) and (5,25)

So, consider the area as a bunch of thin strips, each of width dx, and height equal to the distance between the curves. The area is thus

∫[-4,5] (x+20)-x^2 dx
= 1/2 x^2 + 20x - 1/3 x^3 [-4,5]
= 243/2

How do you get the [-4,5], I don't understand where those numbers come from?

To find the area of the region bounded by the graphs of the given equations, you can follow these steps:

1. Start by graphing the two equations, y = x + 20 and y = x^2, on the same coordinate system. This will give you a visual representation of the region you are looking to find the area of.

2. Find the x-values at which the two graphs intersect. To do this, set the two equations equal to each other and solve for x:

x + 20 = x^2

Rearrange the equation to get a quadratic equation equal to zero:

x^2 - x - 20 = 0

Solve this quadratic equation using factoring, completing the square, or the quadratic formula to obtain the x-values of the points of intersection.

3. Once you have the x-values of the points of intersection, you can find the y-values by substituting these x-values into either of the original equations.

4. Next, determine which equation represents the upper and lower bounds of the region. In this case, y = x^2 is the lower bound and y = x + 20 is the upper bound.

5. Integrate the difference between the upper and lower bounds with respect to x, from the x-value of the leftmost point of intersection to the x-value of the rightmost point of intersection, to find the area:

Area = ∫[lower bound, upper bound] (upper function - lower function) dx

In this case, the integral would be:

Area = ∫[x1, x2] (x + 20 - x^2) dx

Integrate the expression with respect to x within the given limits to obtain the area.

6. Evaluate the integral to find the numerical value of the area.

Following these steps will allow you to find the area of the region bounded by the given equations.