Food for thought

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## Problem

If you know that the area of the shaded region is 40 square units, which of the following can you evaluate?

• $\int_2^6 (f(x) + 5) \ dx$

• $\int_2^6 (f(x) - 3) \ dx$

• $\int_0^4 f(x + 2) \ dx$

• $\int_2^6 f(x + 2) \ dx$

• $\int_4^8 f(x + 2) \ dx$

• $\int_2^6 -f(x) \ dx$

• $\int_2^6 f(-x) \ dx$

If there are any that you cannot evaluate, what additional information would you need in order to do so?

Can you find the value of the constant $k$ for which

$\int_2^6 (f(x) + k) \ dx = 0$

How does your thinking change if instead you start with this image, in which the shaded area is 8 square units?

Which of the integrals above can you evaluate for the linear function $g(x)$?

• How does this compare to your answer for $f(x)$?

• What makes this function different from $f(x)$?

• What else do you know about $g(x)$?