If the area under the curve of f(x) = 25 - x^2 from x = -4 to x = 0 is estimated using four approximating rectangles and left endpoints, will the estimate be an underestimate or overestimate?

Question

Question options:

1)	Underestimate
2)	Overestimate
3)	The area will be exact
4)	The area cannot be estimated with just four rectangles

Answer 1

We can 1.) find the estimated area using the rectangles,

then 2.) find the actual area using an integral,

and 3.) compare these two areas to determine whether the estimate would be larger or smaller.

1. Using rectangles:
2. Because we are using left endpoints, the first rectangle would begin at x= -4. This is the left-most x-value in the interval [-4,0].
3. The y-value at x= -4 is y= 8.
4. We are using 4 approximating rectangles in this interval, meaning the width is 1.
5. This means the first rectangle has dimensions 8 by 1 and an area of 8.
6. The second rectangle begins at x= -3.
7. The y-value here is y=16.
8. The rectangle is 16 by 1 and has an area of 16.
9. The third rectangle begins at x = -2.
10. The y-value is y= 21 means the area is 21.
11. The fourth rectangle begins at x= -1.
12. The y-value is y= 24 means the area is 24.
13. The sum of the areas of these 4 rectangles is 8+16+21+24= 69 squared units.
14. The actual area under f(x) on the interval [-4,0] is given by the integral ∫_-4⁰ 25 - x² dx.
15. Finding the antiderivative, this integral is equal to 25x -x³/3 evaluated from -4 to 0.
16. Substituting final minus initial: 0 - (25(-4) - (-4)³ /3)
17. Evaluating: 0 - (-100 + 64/3) = - (-300 + 64)/3 = 236/3 = 78 and 2/3 squared units
18. The estimated area of 69 is smaller than the actual area of 78 and 2/3. Therefore, the estimate would be an underestimate.