Approximate the area under f(x)=x^2+1 on [2,10] using the areas of rectangles, the right endpoints, and n=4.
and
Show how to determine the value of the actual area under f(x)=x^2+1 on [2,10] by using the fundamental theorem of calculus
and
Show how to determine the value of the actual area under f(x)=x^2+1 on [2,10] by using the fundamental theorem of calculus
-
For the approximation, you are using n=4 to cover the distance from 2 to 10 (8 units), so each of the rectangles will be 2 units wide. To find the height you are asked to use the right endpoints, which will be 4, 6, 8 and 10. For each of these you need f(x) to find the height of the rectangle, so you get 17, 37, 65 and 101.
That means your approximate for the area under the curve is (2*17+(2*37)+(2*65)+(2*101) = 440 square units.
For the exact value, you will simply do the integral and substitute in the values:
f(x)=x^2+1 gives 1/3 x^3 + x when it is integrated. Evaluate it at 10 and 2, and subtract one from the other:
(1/3 (10^3) + 10) - (1/3 (2^3) + 2) = (1/3 (1000) + 10) - (1/3 (8) +2) = 343.33 - 4.6667 = 338.66
You see that your approximation was too large, but this makes sense because the function was increasing and you used the right endpoints, and not very many 'stripes'. Using n=40 would have given a much better approximation, and using the centres instead of the endpoints an even better one.
That means your approximate for the area under the curve is (2*17+(2*37)+(2*65)+(2*101) = 440 square units.
For the exact value, you will simply do the integral and substitute in the values:
f(x)=x^2+1 gives 1/3 x^3 + x when it is integrated. Evaluate it at 10 and 2, and subtract one from the other:
(1/3 (10^3) + 10) - (1/3 (2^3) + 2) = (1/3 (1000) + 10) - (1/3 (8) +2) = 343.33 - 4.6667 = 338.66
You see that your approximation was too large, but this makes sense because the function was increasing and you used the right endpoints, and not very many 'stripes'. Using n=40 would have given a much better approximation, and using the centres instead of the endpoints an even better one.
-
x f(x) left point Right point mid-point integral
2 5 10
3 10 20
4 17 34 34
5 26 52
6 37 74 74
7 50 100
8 65 130 130
9 82 164
10 101 202
using left endpoint with n=4 the answer is 248
using right end point answer is 440
using mid-point answer is 336
integral the answer is 338.7
2 5 10
3 10 20
4 17 34 34
5 26 52
6 37 74 74
7 50 100
8 65 130 130
9 82 164
10 101 202
using left endpoint with n=4 the answer is 248
using right end point answer is 440
using mid-point answer is 336
integral the answer is 338.7