Determine the area of the region bounded by the given function, the x-axis, and the given vertical lines. The region lies above the x-axis.
f(x)=4x√ , x=1 and x=4
a = ∫[1,4] 4√x dx
Now just plug and chug
To determine the area of the region bounded by the function f(x), the x-axis, and the vertical lines x = 1 and x = 4, we need to integrate the function f(x) between those two vertical lines.
f(x) = 4x√
To find the area, we integrate f(x) with respect to x, from x = 1 to x = 4.
∫[1 to 4] 4x√ dx
To evaluate the integral, we use the power rule to integrate the function.
∫[1 to 4] 4x√ dx = [(2/3)x^(3/2)] between 1 and 4
Now, we substitute the upper limit and lower limit into the integral:
[(2/3)(4)^(3/2)] - [(2/3)(1)^(3/2)]
Simplifying:
[(2/3)(8)] - [(2/3)(1)]
(16/3) - (2/3)
= 14/3
Therefore, the area of the region bounded by the function f(x), the x-axis, and the vertical lines x = 1 and x = 4 is 14/3 square units.
To determine the area of the region bounded by the function, the x-axis, and the vertical lines x=1 and x=4, we need to find the definite integral of the function within the given interval.
The definite integral represents the area under the curve of the function between the specified bounds. In this case, we are interested in the area above the x-axis, so we will be looking for the positive area.
To find the area, we can use the definite integral formula:
Area = ∫[a, b] f(x) dx
where f(x) is the given function, and a and b are the lower and upper limits of integration, respectively.
Given that f(x) = 4x√, and the bounds are x=1 and x=4, we have:
Area = ∫[1, 4] 4x√ dx
To find the antiderivative of 4x√, we can use the power rule for integration.
Applying the power rule, we get:
Area = [4/3 * (x^3/2)] evaluated from 1 to 4
Evaluating the integral at the upper and lower limits, we obtain:
Area = [4/3 * (4^(3/2))] - [4/3 * (1^(3/2))]
Simplifying the expression, we have:
Area = [4/3 * 8] - [4/3 * 1]
Area = 32/3 - 4/3
Area = 28/3
Therefore, the area of the region bounded by the function f(x) = 4x√, the x-axis, and the vertical lines x=1 and x=4 is 28/3 square units.