Given the following definite integral of the function f(x)=3x2−2xf\left(x\right)=3x^2-2xf(x)=3x2−2x, write the simplified integral: -∫40f(x)dx\int_4^0\!f\left(x\right)\,dx∫40f(x)dx

3 ∫ 0 4 ⁣ x 2 d x − 2 ∫ 0 4 ⁣ x d x 3\int_0^4\!x^2\,dx-2\int_0^4\!x\,dx 3 ∫ 0 4 x 2 d x − 2 ∫ 0 4 x d x

Given the following definite integral of the function f ( x ) = 3 x 2 − 2 x f\left(x\right)=3x^2-2x f ( x ) = 3 x 2 − 2 x , write the simplified integral: - ∫ 4 0 f ( x ) d x \int_4^0\!f\left(x\right)\,dx ∫ 4 0 f ( x ) d x

in this problem we are told given the following definite integral of this function right here, write the simplified integral of this form. Now what this is basically telling us to do is taking the integral that we have to use all the rules that we know about for integration to write this in the most straightforward way that we could go about solving the problem. So let's see how we can go about doing this. Now one of the first rules I'm going to apply is to notice something here. I noticed we have this negative sign out front, and we also have this kind of weird looking situation where we have a high bound that shows up on bottom and then or a higher number that shows up in the bottom bound and a lower number that shows up in the upper bound. So what I'm gonna do is actually flip these two bounds, and we learned that when we do that, we can actually change the sign of the integral. So we get a negative out front, which will cancel with that negative, meaning we can just write this as the positive integral from zero to four of this function right here. So we have three x squared minus two x. So notice doing that step right there just made this integral look a lot more simple. Now it turns out there are still other ways we can simplify this. Notice we have two portions or two values here that are being added or subtracted. And so what I can do is split this into two separate integrals. This is going to be separately the integral from zero to four of three x squared with respect to x minus, and then we'll have the integral from zero to four of two x with respect to x. Now one last step that I can take is the constant multiple rule, which is I can take any constants we have in front of the x's and I can factor them out. So we'll first, and and we'll start from this left side here, we're first going to take this three and factor that. So we're gonna have three, that's gonna be times the integral from zero to four of our function x squared with respect to x, then we're going to have minus, and then I'll pull this two out, so we'll have two, it's gonna be multiplied by the integral from zero to four of x with respect to x. And this right here is about the most simplified that we could write this integral. So this is how you can solve these types of problems where you need to write the simplified integral and apply the rules that we know about for integration. Hope you found this video helpful, and let's move on.

Given the following definite integral of the function f ( x ) = 3 x 2 − 2 x f\left(x\right)=3x^2-2x f ( x ) = 3 x 2 − 2 x , write the simplified integral: - ∫ 4 0 ⁣ f ( x ) d x \int_4^0\!f\left(x\right)\,dx ∫ 4 0 f ( x ) d x

3 ∫ 0 4 ⁣ x 2 d x − 2 ∫ 0 4 ⁣ x d x 3\int_0^4\!x^2\,dx-2\int_0^4\!x\,dx 3 ∫ 0 4 x 2 d x − 2 ∫ 0 4 x d x

2 ∫ 4 0 ⁣ x d x − 3 ∫ 4 0 ⁣ x 2 d x 2\int_4^0\!x\,dx-3\int_4^0\!x^2\,dx 2 ∫ 4 0 x d x − 3 ∫ 4 0 x 2 d x

3 ∫ 4 0 ⁣ x 2 d x − 2 ∫ 4 0 ⁣ x d x 3\int_4^0\!x^2\,dx-2\int_4^0\!x\,dx 3 ∫ 4 0 x 2 d x − 2 ∫ 4 0 x d x

2 ∫ 0 4 ⁣ x d x − 3 ∫ 0 4 ⁣ x 2 d x 2\int_0^4\!x\,dx-3\int_0^4\!x^2\,dx 2 ∫ 0 4 x d x − 3 ∫ 0 4 x 2 d x

8. Definite Integrals

Introduction to Definite Integrals

Business Calculus

8. Definite Integrals

Introduction to Definite Integrals

Struggling with Business Calculus?

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Multiple Choice

Given the following definite integral of the function $f\left(x\right)=3x^2-2x$ , write the simplified integral:
- $\int_4^0\!f\left(x\right)\,dx$

$2\int_4^0\!x\,dx-3\int_4^0\!x^2\,dx$

$3\int_4^0\!x^2\,dx-2\int_4^0\!x\,dx$

$2\int_0^4\!x\,dx-3\int_0^4\!x^2\,dx$

$3\int_0^4\!x^2\,dx-2\int_0^4\!x\,dx$

Verified step by step guidance

Step 1: Understand the problem. The goal is to rewrite the given definite integral of the function f(x) = 3x^2 - 2x in a simplified form. The integral is given as ∫_4^0 f(x) dx, which represents the area under the curve of f(x) from x = 4 to x = 0.

Step 2: Recognize that the integral bounds are reversed (from 4 to 0 instead of 0 to 4). To simplify, use the property of definite integrals: ∫_a^b f(x) dx = -∫_b^a f(x) dx. This means ∫_4^0 f(x) dx = -∫_0^4 f(x) dx.

Step 3: Substitute the function f(x) = 3x^2 - 2x into the integral. This gives -∫_0^4 (3x^2 - 2x) dx. Use the linearity property of integrals to split this into two separate integrals: -[∫_0^4 3x^2 dx - ∫_0^4 2x dx].

Step 4: Factor out the constants from each integral. For the first term, factor out 3 to get -3∫_0^4 x^2 dx. For the second term, factor out 2 to get +2∫_0^4 x dx. This simplifies the expression to -3∫_0^4 x^2 dx + 2∫_0^4 x dx.

Step 5: Rearrange the terms to match the format of the answer choices. This gives 3∫_0^4 x^2 dx - 2∫_0^4 x dx, which is the simplified form of the given integral.

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