Area and Limits

Question One Using geometry, find the area of the shaded region under the curve f(x) = 4.9 from x = 0 to x = 10. Area of a rectangle: Length x Width 10 units x 5 units = 50 units squared.

Question Two Calculate the area of this shaded region by using the area formula for a trapezoid. Area of a trapezoid: (1/2) x Height x (Base 1 + Base 2) (1/2) x 10 units x (5 units + 9 units) = 70 units squared

Question Three This function has the same y-values as the previous linear function at x = 0 and x = 10. How do you think the area of the shaded region will compare to the area of the region on the previous page? The area of the shaded region will be less than the area of the region on the previous page, because it is concave up.

Question Four What is your estimate for the area of the shaded region if you use only whole boxes? Is this an overestimate or an underestimate? There are eleven whole boxes under the curve, so the estimate is 11 units squared. It is an underestimate, because there are still portions of boxes left over.

Question Five What could you do to make a better or more accurate estimate? You could estimate the area of boxes that aren’t whole and add them to your estimate for the entire region.

Question Six Use the y-coordinate of a point C to calculate the area of each of the rectangles. Write down their areas and find the sum. What is your estimate for this case? Is it an underestimate or an overestimate? = 50.5 units squared. This is an underestimate, because the rectangles do not equal the entire amount under the curve.

Question Seven Using right rectangles, what is the value for Area Sum in this case? Given just the expression for the function f, how could you find this sum? Is it an underestimate or an overestimate? = 54.5 units squared. Give just the expression, you could find this sum by multiplying the base of the rectangles by f(x) at the point, and adding them together. It is an overestimate, because the rectangles go out farther than the curve.

Question Eight Back in steps 5 – 6, you looked at rectangles with a base of 10 units. One gave an underestimate of 49 square units, and the other gave an overestimate of 89 square units – a difference of 40 units. How far apart are your two estimates in Question Six and Question Seven using rectangles with a base of one unit? They are four units apart.

Question Nine What do you think will happen to the difference between the estimates if you use rectangles of width 0.5? The difference between the estimates will become even smaller if you use rectangles with smaller width.

Question Ten What is the exact area of the shaded region under the curve f(x) = x + 1 on the domain [0,10]? The exact area of the shaded region under the curve f(x) = x + 1 on the domain [0,10] is 50 square units.

Question Eleven Write an expression to calculate the area of the left sum with five rectangles. 2f(0) + 2f(2) + 2f(4) + 2f(6) + 2f(8)

Question Twelve How would this expression change if there were six, seven, eight, ten, twenty, or even more rectangles? Write a general expression for the sum for n rectangles using ∑ notation.

Question Thirteen Using this expression, write a limit to express the area of the shaded region.

Question Fourteen Write an expression for the sum of the ten rectangles you made for the function f(x) = 0.1(x-3)^2 + 4 back in Question Seven. How can you find the sum of these rectangles without adding everything up by hand?