Set up a definite integral that yields this region
f(x)=4-2x
Base is 2, and height is 4
do you mean the area under the line? If so, just draw the line, and you can see that it is
∫[0,2] 4-2x dx
Well, I have to say, this is quite the mathematical circus we're about to perform! To find the definite integral that gives the area of the region bounded by the function f(x) = 4 - 2x and the x-axis on the interval [2, b], where the base is 2 and the height is 4, we need to use a little bit of clown math.
Let's start by finding where the function intersects the x-axis, because that will give us the value of b. Setting f(x) = 0, we get:
4 - 2x = 0
2x = 4
x = 2
Since the base is 2 units, our interval starts at x = 2. And since the height is 4 units, our function f(x) = 4 - 2x establishes the top boundary of our region. So, the definite integral that gives the area of this region is:
∫[2,b] (4 - 2x) dx
But since we still need to find the upper limit b, we will have to integrate this expression after finding the value of b. Hence, our mathematical circus will continue!
To set up a definite integral that yields the region bounded by the function f(x) = 4 - 2x, the base of 2, and the height of 4, you first need to determine the limits of integration.
Since the base is given as 2, the integral will be evaluated over the interval [a, b], where a = 0 and b = 2.
To find the height of the region, observe that the function f(x) = 4 - 2x reaches a maximum value of 4 at x = 0 and decreases linearly to a minimum value of 0 at x = 2. Therefore, the height is given by h(x) = (4 - 2x).
The area of the region can be represented as the integral of the height function h(x) over the interval [a, b]:
∫[a, b] h(x) dx = ∫[0, 2] (4 - 2x) dx
Now, you can evaluate this definite integral to find the area of the region bounded by the given function, base, and height.
To set up a definite integral that yields the given region, we need to calculate the area under the curve defined by the function f(x) = 4 - 2x between the x-values of the base and a horizontal line that corresponds to the height.
Given that the base is 2 units and the height is 4 units, we need to find the x-values where the curve intersects the horizontal line at y = 4.
To find the x-values, we set f(x) = 4 - 2x equal to 4 and solve for x:
4 - 2x = 4
-2x = 0
x = 0
So, the curve intersects the line y = 4 at x = 0.
Now we can set up the definite integral to calculate the area under the curve between x = 0 and x = 2.
∫[0, 2] (4 - 2x) dx
To evaluate this integral, we can integrate the function 4 - 2x with respect to x:
∫[0, 2] (4 - 2x) dx = [4x - x^2] evaluated from 0 to 2
= (4(2) - 2^2) - (4(0) - 0^2)
= (8 - 4) - (0 - 0)
= 4
Therefore, the definite integral ∫[0, 2] (4 - 2x) dx yields an area of 4, which represents the region under the curve f(x) = 4 - 2x, with the base of 2 units and height of 4 units.