calculate the upper limit for the definite integral from -2 to 2. oh x^2+1 dx with the partition P=-2,-1,0,1,2
To calculate the upper limit of the definite integral, we need to first find the upper sum of the given function.
The upper sum is calculated by taking the maximum value of the function within each subinterval of the partition and multiplying it by the width of that subinterval. Then, we sum up all these products.
Let's follow these steps to find the upper limit:
1. Calculate the width of each subinterval:
Δx₁ = -1 - (-2) = 1
Δx₂ = 0 - (-1) = 1
Δx₃ = 1 - 0 = 1
Δx₄ = 2 - 1 = 1
2. Find the maximum value of the function within each subinterval:
For the subinterval [-2, -1]:
f(-2) = (-2)² + 1 = 4 + 1 = 5
For the subinterval [-1, 0]:
f(-1) = (-1)² + 1 = 1 + 1 = 2
For the subinterval [0, 1]:
f(0) = (0)² + 1 = 0 + 1 = 1
For the subinterval [1, 2]:
f(2) = (2)² + 1 = 4 + 1 = 5
3. Calculate the upper sum:
Upper Sum = (5 * 1) + (2 * 1) + (1 * 1) + (5 * 1)
= 5 + 2 + 1 + 5
= 13
Therefore, the upper limit for the definite integral from -2 to 2 of the function f(x) = x^2 + 1 using the given partition P = {-2, -1, 0, 1, 2} is 13.