Evaluate the integral by interpreting it in terms of areas.
int_(0)^(9) (7/2x - 21)dx
can someone please help me with this question i keep on getting the wrong answer...the answer i got was -324/4
The integral is (7/4 x^2-21x) evaluated from zero to 9 is 7*81/4-21(9) How did you get your answer?
so the answer is 567/-185
To evaluate the integral ∫[0, 9] (7/2x - 21)dx by interpreting it in terms of areas, we can break it down into two separate integrals and consider each one individually.
The given integral is:
∫[0, 9] (7/2x - 21) dx
We can rewrite this as:
∫[0, 9] (7/2x) dx - ∫[0, 9] 21 dx
Now let's evaluate each integral separately.
1. ∫[0, 9] (7/2x) dx:
To find the integral of (7/2x) with respect to x, we can rewrite it as:
(7/2) ∫[0, 9] x dx
Using the power rule for integration, we have:
(7/2) * (1/2) * x^2 evaluated from x = 0 to x = 9:
(7/2) * (1/2) * (9^2) - (7/2) * (1/2) * (0^2)
= (7/2) * (1/2) * 81 - (7/2) * (1/2) * 0
= (7/2) * (1/2) * 81
= 567/4
2. ∫[0, 9] 21 dx:
To find the integral of 21 with respect to x, we can rewrite it as:
21 ∫[0, 9] dx
Using the definite integral property, the integral of a constant is the constant multiplied by the length over which the integration is being performed:
21 * (9 - 0) = 21 * 9 = 189
Now, subtract the two integrals:
(567/4) - 189
To compute this subtraction, we need to find a common denominator:
(567/4) - (756/4)
Now subtract the numerators:
(567 - 756)/4
= (-189)/4
= -189/4
Therefore, the value of the integral ∫[0, 9] (7/2x - 21) dx is -189/4.
To evaluate the integral by interpreting it in terms of areas, you need to understand that integrating a function represents finding the area between the function and the x-axis over a given interval. In this case, you want to evaluate the integral of the given function f(x) = (7/2x - 21) over the interval [0, 9].
To start, let's break down the integral into two separate parts:
∫[0, 9] (7/2x - 21) dx = ∫[0, 9] (7/2x) dx - ∫[0, 9] 21 dx
Now, let's evaluate each integral separately:
1. ∫[0, 9] (7/2x) dx:
To integrate (7/2x), we can use the power rule of integration, which states that ∫x^n dx = (1/(n+1)) * x^(n+1). Applying this rule, we get:
∫(7/2x) dx = (7/2) * (1/2) * x^2 = (7/4) x^2
Now, we calculate the definite integral for this part:
∫[0, 9] (7/2x) dx = [(7/4) * 9^2] - [(7/4) * 0^2] = (7/4) * 81 = 567/4
2. ∫[0, 9] 21 dx:
Integrating a constant term such as 21 with respect to x yields:
∫21 dx = 21x
Evaluating the definite integral for this part:
∫[0, 9] 21 dx = 21*9 - 21*0 = 189
Finally, we add the results of both integrals to get the overall area:
∫[0, 9] (7/2x - 21) dx = ∫[0, 9] (7/2x) dx - ∫[0, 9] 21 dx = (567/4) - 189 = 567/4 - 189 = (567 - 4*189)/4 = (567 - 756)/4 = -189/4
Therefore, the correct answer is -189/4. It seems there was an error in your calculation.