find the area of the region between the curve y = 3-x^2 and the line y = -1 by integrating with respect to y
you first have to solve for x
x=(3-y)^(1/2)
then area
⌠ (3-y)^(1/2 dy from -1 to 3
⌡
=-2/3 (3-y)^(3/2 | from -1 to 3
= -2/3(0 - 8)
= 16/3
but this is only half the region, so the area = 32/3 units^2
Question: Why would anybody ask you to do a question the hard way, the other way with respect to x is easier.
That's a valid question. Sometimes, problems are presented in different ways to practice different techniques or to challenge students to think critically. In this case, the problem asked you to find the area between a curve and a line by integrating with respect to y, rather than x. It might not be the easiest approach, but it offers an opportunity to practice working with different methods and gain a deeper understanding of integration.
The alternative method, integrating with respect to x, might be simpler in this case, as you mentioned. It involves solving for y in terms of x and then finding the area between the two curves using the limits of integration as the x-values where they intersect.
So, while integrating with respect to y might not be the most efficient approach in this particular problem, it's important to be flexible and able to solve problems using various methods to gain a comprehensive understanding of the topic.