Question

Evaluate the integral from 0 to 10 of the function f of x, dx for f of x equals 5 for x less than or equal to 5 and equals the quantity 10 minus x for x greater than 5.

(A) 37.5
(B) 25
(C) 12.5
(D) 50

Answer 1

To evaluate the integral from 0 to 10 of the function f(x), we split the interval into two parts. The value of the integral is 75.

Step-by-step explanation:

To evaluate the integral from 0 to 10 of the given function f(x), we need to split the interval into two parts. From 0 to 5, the function is constant at f(x) = 5. From 5 to 10, the function is f(x) = 10 - x. Let's calculate the integral:

∫010 f(x) dx = ∫05 5 dx + ∫510 (10 - x) dx

Using basic integral rules, the first integral evaluates to 5 * x, while the second integral evaluates to 10x - x^2/2.

Now, we substitute the limits of integration and calculate the integral:

∫05 5 dx + ∫510 (10 - x) dx = (5 * 5) + [(10 * 10) - (10^2/2)] = 25 + [(100) - (50)] = 25 + 50 = 75.

Therefore, the value of the integral from 0 to 10 of the function f(x) is 75.