Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval.

Question

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f(x) = x⁶, [0, 6]

Answer 1

c=f^-1(area/interval) where area equal integral of f(x) over interval

f(x)=x⁶ integral is x⁷/7

area x=0 to x=6 is,,,,6⁷/7

area divided by interval is 6⁶/7

f^-1(x)=x^(1/6)

c=(6⁶/7)^(1/6)=6(1/7)^(1/6)=6*.723

c=4.338

check work, 4.338⁶=6664. is the f(x) value corresponding to c ,, f(c)

6664*6=39984.2 is value times interval,,,,,f(c)*(b-a)

6⁷/7=39990.9

Answer 2

We're seeing the words "Mean Value Theorem" here. Recall that we've seen this before when learning about the derivative of a function f(x). But here, we're talking about the Mean Value Theorem for Integrals, which states the following:

If f(x) is continuous on a closed interval [a,b], then there exists at least one point, call it c, such that c is in the interval [a,b], that is, c is a number between a and b, such that ∫_a^b f(x) dx = f(c) ( b - a ).

Let's unpack this! The left hand sides ∫_a^b f(x) dx will come out to some number. The right hand side f(c) (b-a) will be come expression involving c. This means that we need to solve for c.

Our function is f(x) = x⁶ on the interval [0,6]. Okay! Let's first start by finding this value by using the Fundamental Theorem of Calculus!

∫₀⁶ x⁶ dx = [ x⁷ / 7 ] ₀⁶ = [ (6)⁷ / 7 ] - [ (0)⁷ / 7 ] = 39990.9 - 0 = 39990.9

Fantastic! We've got our number! Note we need to compute f(c)(b-a) = f(c) (6 - 0) = 6c⁶. The mean value theorem says that we can solve the equation

6c⁶ = 39990.9

for c! So let's do that!

6c⁶ = 39990.9

⇒ c⁶ = 6665.14 (must take the 6-th root of both sides)

⇒ c = 4.33812

Now! We're not quite finished! Notice that this mean value theorem says we need our solutions for c to be in the interval [0,6]. Since 4.33812 is between 0 and 6, we know that c = 4.33812 is out solution and we're done!