Video: Expressing the Sum of Three Definite Integrals over Adjacent Intervals as One Integral

Write ∫_(−2) ^(3) (𝑓(𝑥) d𝑥) + ∫_(3) ^(4) (𝑓(𝑥) d𝑥) − ∫_(−2) ^(0) (𝑓(𝑥) d𝑥) in the form ∫_(𝑎) ^(𝑏) (𝑓(𝑥) d𝑥).

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Video Transcript

Write the integral from negative two to three of 𝑓 of 𝑥 with respect to 𝑥 plus the integral from three to four of 𝑓 of 𝑥 with respect to 𝑥 minus the integral from negative two to zero of 𝑓 of 𝑥 with respect to 𝑥 in the form of the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 with respect to 𝑥.

We know that if capital 𝑓 of 𝑥 is the antiderivative of 𝑓 of 𝑥, then the integral from a lower bound 𝑙 to an upper bound 𝑢 of 𝑓 of 𝑥 with respect to 𝑥 is equal to evaluating our antiderivative at 𝑢 and subtracting this from our antiderivative evaluated at 𝑙. We can use this to rewrite the integrals given to us in this question. We can start by rewriting the integral from negative two to three of 𝑓 of 𝑥 with respect to 𝑥 as capital 𝐹 of three minus capital 𝐹 of negative two. We can do the same with the integral from three to four of 𝑓 of 𝑥, giving us capital 𝐹 for four minus capital 𝐹 of three. Finally, we can rewrite the last integral in our question of 𝑓 of 𝑥 from negative two to zero with respect to 𝑥 as capital 𝐹 of zero minus capital 𝐹 of negative two.

We can now substitute these values into the expression given to us in the question. We rewrite the first term as capital 𝐹 of three minus capital 𝐹 of negative two. We then add on our new expression for a second term, which is capital 𝐹 of four minus capital 𝐹 of three. Finally, we can change the last expression given to us in this question with capital 𝐹 of zero minus capital 𝐹 of negative two. Well, we remember to subtract this expression because the integral given to us in the question is negative.

Now, we just expand the brackets given to us in our new expression. We can simplify this expression by canceling out terms, giving us that the expression given to us in the question is equal to evaluating the antiderivative of 𝐹 of four and subtracting the antiderivative of 𝐹 evaluated at zero. Finally, since capital 𝐹 is the antiderivative of 𝑓, we can use our same rule to rewrite this expression as the integral from zero to four of 𝑓 of 𝑥 with respect to 𝑥.

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