Consider the area between the graphs x+3y=35 and x+5=y2. This area can be computed in two different ways using integrals

probably true. I'll show you mine if you show me yours...

To compute the area between two curves using integrals, you can follow these steps:

Step 1: Find the points of intersection.
To compute the area between the graphs x + 3y = 35 and x + 5 = y^2, you need to find the points of intersection, where the two curves intersect.

Using substitution method, solve for x in terms of y for each equation:
For x + 3y = 35, rearrange the equation to solve for x:
x = 35 - 3y.

For x + 5 = y^2, rearrange the equation to solve for x:
x = y^2 - 5.

Now set these two equations equal to each other to find the points of intersection:
35 - 3y = y^2 - 5.

Simplifying the equation gives: y^2 + 3y - 40 = 0.

Step 2: Find the x-values of the points of intersection.
To find the x-values of the points of intersection, substitute the y-values found in Step 1 into either of the original equations (x + 3y = 35 or x + 5 = y^2) and solve for x.

For example, using x = 35 - 3y, substitute the y-values found in Step 1 to obtain the corresponding x-values.

Step 3: Set up the integral(s).
Next, determine which curve is on top by comparing their y-values at each point of intersection. The curve with the larger y-value at any given x will be considered "on top" between those x-values.

If the region has multiple sections, split the area into separate integrals for each section.

Step 4: Evaluate the integral(s).
Using the appropriate integral(s) for each section, integrate the difference between the upper curve and lower curve with respect to x. The resulting integral(s) will give you the area between the curves.

By following these steps, you can compute the area between the two given curves using integrals.