Answer:
≈ 562.5
Explanation:
To estimate the area under the curve and above the x-axis with 4 subintervals, we can use the trapezoidal rule.
First, we need to find the width of each subinterval. Since we are using 4 subintervals, the width will be:
Δt = (b - a) / n = (20 - 0) / 4 = 5
where a = 0 and b = 20 are the limits of integration, and n is the number of subintervals.
Next, we need to evaluate the function at the endpoints of each subinterval:
F(0) = 0
F(5) = 5(20 - 5) = 75
F(10) = 10(20 - 10) = 100
F(15) = 15(20 - 15) = 75
F(20) = 20(20 - 20) = 0
Then, we can use the trapezoidal rule formula:
∫[a,b]f(x)dx ≈ (Δt / 2) [f(a) + 2f(a + Δt) + 2f(a + 2Δt) + 2f(a + 3Δt) + f(b)]
Plugging in the values we just calculated, we get:
∫[0,20]t(20-t)dt ≈ (5 / 2) [0 + 2(75) + 2(100) + 2(75) + 0]
≈ 562.5
Therefore, the estimated area under the curve and above the x-axis with 4 subintervals is approximately 562.5 square units.