Consider the integral:
21
∫ x^3 dx
3
-Use all three methods to approximate this integral with n = 3
-Find the exact value of this integral
I have no way of knowing what "all three methods" are, I will just
do the exact value method, that is , using the regular integration method
21
∫ x^3 dx
3
= [ (1/4)x^4] from 3 to 21
= ( (1/4)(21^4) - (1/4)(3^4) )
= (1/4)(194481 - 81)
= 194400/4 = 48600
three methods?
Left Sum, Right Sum, Midvalue Sum, Trapezoid Rule ...
Just break it up into 3 intervals of width 6 and apply the formulas.
To approximate the integral using three methods with n = 3, we can use the Left Riemann Sum, the Right Riemann Sum, and the Midpoint Rule.
1. Left Riemann Sum:
To approximate the integral using the Left Riemann Sum, we divide the interval [21, 3] into n = 3 subintervals of equal width. In this case, each subinterval width is (3-21)/3 = 6. Then, we evaluate the function at the left endpoint of each subinterval and multiply it by the width of the subinterval. Finally, we sum up all these products.
Approximation using Left Riemann Sum = (f(21) * 6) + (f(15) * 6) + (f(9) * 6)
2. Right Riemann Sum:
Similar to the Left Riemann Sum, we divide the interval [21, 3] into n = 3 subintervals of equal width. However, this time we evaluate the function at the right endpoint of each subinterval.
Approximation using Right Riemann Sum = (f(27) * 6) + (f(21) * 6) + (f(15) * 6)
3. Midpoint Rule:
Again, we divide the interval [21, 3] into n = 3 subintervals of equal width, but this time we evaluate the function at the midpoint of each subinterval and multiply by the width.
Approximation using Midpoint Rule = (f(18) * 6) + (f(12) * 6) + (f(6) * 6)
To find the exact value of the integral, we can calculate the indefinite integral of x^3:
∫ x^3 dx = (1/4) * x^4 + c
Now, we can use the Fundamental Theorem of Calculus to evaluate the definite integral.
Exact value of the integral = F(3) - F(21)
where F(x) = (1/4) * x^4 + c
Now, you can substitute the upper and lower limits into the function F(x) and calculate the exact value.