Use geometry to 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 .
12.5
25
37.5
Cannot be found
Steve you are an !@#$%^&
Answer is 37.5
It is 37.5.
The geometric representation of an integral is the area below the curve. So, let's graphic the function.
F(x) = 5 is an horizontal line, (y = 5)
and f(x)=10-x is also a line, but it is decreasing.
If you graph this out, you can see in the image that the area below the curve happens to be a polygon. In order to calculate the area we are going to separate it into simpler polygons: a square and a triangle.
The area of the square is 5 times 5 = 25.
The area of the triangle is 5 times 5 divided by 2 = 12.5.
If we add them, we get the integral from 0 to 10 of the function f of x, dx = 25 + 12.5 = 37.5
Well, evaluating this integral is going to be as fun as trying to fit a square peg in a round hole! Let's break it down into two parts, just like a rhombus that has become the class clown.
First, let's integrate the function when x is less than or equal to 5. Since f(x) is equal to 5 in this range, the integral from 0 to 5 of 5 dx is simply 5 times the integral of 1, which is just 5x evaluated from 0 to 5. Plugging in the limits, we get 5(5) - 5(0) = 25.
Next, let's integrate the function when x is greater than 5. The function becomes f(x) = 10 - x in this range. So, the integral from 5 to 10 of (10 - x) dx is equal to the integral of 10 dx minus the integral of x dx, evaluated from 5 to 10. Integrating 10 dx gives us 10x, while integrating x dx gives us (1/2)x^2. Evaluating the integrals from 5 to 10, we get 10(10) - (1/2)(10)^2 - [10(5) - (1/2)(5)^2] = 100 - 50 - (50 - 12.5) = 12.5.
Now, let's add the two parts together. 25 + 12.5 = 37.5. So, the value of the integral from 0 to 10 of f(x) dx is 37.5. Just like a geometry question, this integral was all about fitting the pieces together perfectly!
To evaluate the integral of the function f(x) over the interval [0, 10], we need to split the integral into two parts based on the given definition of f(x).
For x values less than or equal to 5, the function is f(x) = 5. Therefore, the integral on this interval is:
∫[0,5] 5 dx
Since the function is constant, integrating it will simply give us the product of the function value and the width of the interval:
∫[0,5] 5 dx = 5 * (5 - 0) = 5 * 5 = 25
For x values greater than 5, the function is defined as f(x) = 10 - x. So, the integral on this interval is:
∫[5,10] (10 - x) dx
We can integrate this using the power rule:
∫[5,10] (10 - x) dx = (10x - (1/2)x²) | [5,10]
= [(10*10 - (1/2)*10²) - (10*5 - (1/2)*5²)]
= [(100 - 50) - (50 - 12.5)]
= (50 - 37.5)
= 12.5
Therefore, the total value of the integral from 0 to 10 is the sum of the two evaluated integrals:
∫[0,10] f(x) dx = ∫[0,5] 5 dx + ∫[5,10] (10 - x) dx
= 25 + 12.5
= 37.5
Thus, the value of the integral from 0 to 10 of the given function f(x) is 37.5. Therefore, the correct answer is 37.5.