Find the area of the region bounded by the parabola the tangent line to this parabola at (1, 1) that you found in (a), and the
x-axis.
The tangent line is y= 2x-1.
Here's a graph of what it looks like..
I can't seem to find the right answer. I found the integral of x^2-2x+1 from 0 to 1, and get the answer 1/3.
Alternatively, calculate the area under the curve from x=0 to x=1.
Then subtract the area of the half-rectangle from x=0.5 to x=1
The integral will include the area under the line and above the x-axis from x=0.5 to x=1, hence we must subtract the area under the line in order to find the area between the line and curve, bounded by the positive x and y axes, or take the route shown by Von Nemo.
Let's do it with respect to y, It'll be easier.
Recall that
when we want to find area between curves with respect to y.
Step 1. Find limits of integration.
We need to know over what interval y will move along. This is easy. From the context of the question (i.e find the are between curves from y=0 to y=1).
So, we see that and .
Step 2 Determine which is the and which is the .
If we were to draw a line which is parallel to the x-axis we would see that the curve would intersct this line to the left of where would intersect said line. Therefore and .
Step 3. Get every thing in terms of y
And
Step 4. Substitute the relevant information into the formula
Step 5. Evaluate
If you need help with step 5, let me know.