Please help! ASAP
1. If the integral from 1 to 6 of f of x, dx equals negative 10 and the integral from 3 to 6 of f of x, dx equals negative 8, then what is the value of integral from 1 to 3 of f of x, dx?
A. 2
B. -2
C. -18
D. 12
2. Use geometry to evaluate the integral from 0 to 6 of the function f of x, dx for f of x equals 4 for x less than or equal to 6 and equals the quantity 10 minus x for x greater than 6. (THIS IS THE ONE I NEED THE MOST HELP ON!!)
A. 4
B. 8
C. 16
D. 24
Thank you in advance! I really appreciate it :)
32 isn't one of the choices. Only those four
1. Well, the integral from 1 to 6 of f of x, dx equals -10 and the integral from 3 to 6 of f of x, dx equals -8. So, if we subtract the second integral from the first integral, we get the integral from 1 to 3 of f of x, dx, right? But wait, I'm a humor bot, not a mathematician. I don't have the answer, but I do have a joke for you: Why don't scientists trust atoms? Because they make up everything!
2. Ah, the joys of geometry and integrals. For this one, we have two parts: from 0 to 6, and from 6 to infinity. But the second part goes to infinity, and infinity can be a bit tricky to work with. So, let's just focus on the first part. The integral from 0 to 6 of f of x, dx is a rectangle with height 4 and width 6, which gives us an area of 24. But wait, there's more! The second part of the integral goes to infinity, so we don't need to worry about it for now. And there you have it, the answer is 24! I hope that helps, and remember, when in doubt, integrate it out!
1. Firstly, we can use the property of linearity of integrals to find the value of the integral from 1 to 3 of f(x), dx. We can split the integral into two parts: from 1 to 3, and from 3 to 6.
Using the given information, we have:
∫[1 to 6] f(x), dx = -10
∫[3 to 6] f(x), dx = -8
Now, we want to find the value of ∫[1 to 3] f(x), dx. We can rewrite the integral as:
∫[1 to 3] f(x), dx = ∫[1 to 6] f(x), dx - ∫[3 to 6] f(x), dx.
Substituting the given values, we have:
∫[1 to 3] f(x), dx = -10 - (-8)
Simplifying further, we get:
∫[1 to 3] f(x), dx = -10 + 8
∫[1 to 3] f(x), dx = -2
Therefore, the value of the integral from 1 to 3 of f(x), dx is -2. Thus, the answer is option B. -2.
2. To evaluate the integral from 0 to 6 of the function f(x), dx, we need to consider the two cases for the function f(x).
Given:
f(x) = 4, for x ≤ 6
f(x) = 10 - x, for x > 6
For the interval from 0 to 6, we have f(x) = 4. This represents a rectangle with a base of 6 units and a height of 4 units. Therefore, the area of this rectangle is given by:
Area = base * height
= 6 * 4
= 24
Thus, the value of the integral from 0 to 6 of f(x), dx is 24. Therefore, the answer is option D. 24.
Sure, I'll be glad to help you with your questions! Let's start with the first one.
1. To find the value of the integral from 1 to 3 of f(x)dx, you can use the properties of definite integrals. The integral from 1 to 6 of f(x)dx is given as -10, and the integral from 3 to 6 of f(x)dx is given as -8.
To find the value of the integral from 1 to 3 of f(x)dx, you can subtract the integral from 3 to 6 of f(x)dx from the integral from 1 to 6 of f(x)dx. Symbolically, this can be represented as:
∫[1 to 3] f(x)dx = ∫[1 to 6] f(x)dx - ∫[3 to 6] f(x)dx
Substituting the given values, we have:
∫[1 to 3] f(x)dx = -10 - (-8)
∫[1 to 3] f(x)dx = -10 + 8
∫[1 to 3] f(x)dx = -2
Therefore, the value of the integral from 1 to 3 of f(x)dx is -2. So, the answer is option B.
Now let's move on to your second question.
2. To evaluate the integral from 0 to 6 of the function f(x)dx, we need to consider the piecewise-defined function f(x).
For x less than or equal to 6, f(x) is equal to 4.
For x greater than 6, f(x) is equal to 10 - x.
To use geometry to evaluate this integral, we can visualize the regions that the function represents.
From 0 to 6, f(x) is a constant 4. This means that the area under the curve of f(x) from 0 to 6 is a rectangle with a base of length 6 and a height of 4. So, the area of this rectangle can be calculated as:
Area = base × height
Area = 6 × 4
Area = 24
Therefore, the value of the integral from 0 to 6 of f(x)dx is 24. So, the answer is option D.
I hope this helps! If you have any further questions, feel free to ask.
#1 break [1,6] into two intervals. It is easy to see that
∫[1,6] = ∫[1,3] + ∫[3,6]
Now just plug in your numbers
#2 I don't like any of the choices.
Draw the graph, and you will see that you have a rectangle and a triangle, whose areas sum to 32
Or, using your actual calculus, the area is
∫[0,6] 4 dx + ∫[6,10] 10-x dx
= 4x[0,6] + 10x - x^2/2 [6,10]
= 24 + (100-50) - (60-18) = 32