Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 . &...

Question

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .

c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ?

Accepted Answer

Welcome back, everyone. Consider H of X equals X2 + MX where M is less than 0. For what value of N greater than 0 in terms of M does the integral from 0 ton of HXDX equal 0? For this problem, let's integrate H of X with respect to X. So we're going to integrate X2 plus MXDX. And our limits of integration would be from 0 to N, right? So for so we're integrating X2d is Integral would be X cubed divided by 3. Using the power rule. M is a constant, so we're going to add M multiplied by the integral of X, which is X 2 divided by 2 again using the power rule. And we want to validate this result from 0 up to N, right? Using the evaluation theorem, we're going to get N cubed divided by 3 plus M multiplied by N2d divided by 2. And then we're subtracting this result at 0, which is 0, right, because every term contains X. So we would get 0 cubed divided by 3 plus M multiplied by 0 square divided by 2, which is 0. So that we can show that this is equal ton cubed, divided by 3 plus MN 2 divided by 2. And we want to set it equal to 0. So what we're going to do is simply perform the calculations. What we can do is multiply both sides by 6, because that's the common denominator of 3 and 2. So let's multiply both sides of this equation by 66 divided by 3 is 2, so we get 2 and cubed plus 6 divided by 2 is 3, so we're adding 3 m and squared. And this is equal to 0 multiplied by a 6, which is 0. Now we can factor out m squared. And we get 2 n in C plus 3M. That's equal to 0, right? We know that M must be negative and N must be positive. Considering the solutions of this equation, we can either have n2 is equal to 0, which means that N is equal to 0. But N cannot be zero because it's strictly greater than 0, right? So we're only considering our second factor. In other words, 2N plus 3M is equal to 0. And now we're going to solve for N. 2 N is equal to -3M. We can see that N is going to be greater than 0 if M is going to be negative, right? Because -3, multiplied by a negative number, we turns a positive number. And now dividing both sides by 2, we get N equals -3M, our right hand side divided by 2. That would be our final answer and thank you for watching.

Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .
c) In general, for the function ƒ(𝓍) = 𝓍² ― a𝓍, where a > 0, for what value of b > 0 (as a function of a) is ∫₀ᵇ ƒ(𝓍) d𝓍 = 0 ?

Key Concepts

Definite Integral and Net Area

Finding the Integral of a Quadratic Function

Solving for the Upper Limit to Achieve Zero Net Area

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