Evaluate the following sum sigma^20_i = 1 i: Find the area bounded by the graphs

Question

Evaluate the following sum sigma^20_i = 1 i: Find the area bounded by the graphs of the following two functions y = sin^2 x, 0 lessthanorequalto x lessthanorequalto pi and y = 0.. We know that integral^1_0 1/1 + x^2 dx = pi/4, try to use midpoint rule to the integral with n = 5 to find the approximate value of pi/4. Use the formula 1 + 2^2 + 3^2 +.... +n^2 = sigma^n_i = 1 i^2 = n(n + 1)(2n + 1)/6, to evaluate the following definite integral: integral^1_0 x^2 dx. (Using Newton-Leibniz formula will have no points! And you can choose left-point approximation or right-point approximation freely.)

Explanation / Answer

7.

(1)

[i=1 to n]i =n(n+1)/2

[i=1 to 20]i =20(20+1)/2

[i=1 to 20]i =10*21

[i=1 to 20]i =210

(2)

area bounded by the graphs =[0 to ] [sin2x -0] dx

area bounded by the graphs =[0 to ] [sin2x] dx

area bounded by the graphs =[0 to ] (1/2)[1-cos2x] dx

area bounded by the graphs =[0 to ] (1/2)[x-(1/2)sin2x]

area bounded by the graphs =(1/2)[-(1/2)sin2] -(1/2)[0-(1/2)sin0]

area bounded by the graphs =(1/2)[-0] -(1/2)[0-0]

area bounded by the graphs =(/2) -0

area bounded by the graphs =(/2) square units

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