Thread: Finding area of region bounded (Integration)

Hi, so I have this problem:
To find the area of the region bounded by graphs of y=x³, y=0, x=1, and x=3 by limit definition

But I was a bit confused on this because I didn't really know what the intervals were.
Like I graphed it out, and it would look like this:
I will describe it: A horizontal line on y=0, then 2 veritcal lines on x=1 and x=3, then the x³ curves from down to up, hits the 0, but the shape looks pretty weird.

Can someone explain to me what area/intervals are supposed to be?
From my intepretation when I saw the graph of all those equations, it looked like it would be [1, 3] since 3 looks like the max, and 1 is like the min, and there's where they all come together

They make somewhat like a rectangle at the bottom, then it curves up, so like a curvy trapezoid? Maybe?

I think you have the region right - its boundaries are the vertical lines and on the right and left, the horizontal line on the bottom, and the curve on the top.

The way I learned it was to divide the region into vertical rectangles. These rectangles have width and height , so the sum of is the area of the region. In the limit, it's the integral of and the limits are on the left and on the right. So the area is given by:

.

Hopefully that answers your question.

- Hollywood

Originally Posted by hollywood

I think you have the region right - its boundaries are the vertical lines and on the right and left, the horizontal line on the bottom, and the curve on the top.

The way I learned it was to divide the region into vertical rectangles. These rectangles have width and height , so the sum of is the area of the region. In the limit, it's the integral of and the limits are on the left and on the right. So the area is given by:

.

Hopefully that answers your question.

- Hollywood

Ooo
I see.
So the two x's are the boundaries, and the slope is x^3
Not sure where the y=0 goes.
But I think I get it, thanks