Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.




∫₀ᵃ ƒ(𝓍) d𝓍

Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.                                                                                                                                      

 ∫₀⁴ (8―2𝓍) d𝓍

Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = cos 𝓍 on [―π/2 , π/2]

If ƒ is an odd function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 0?

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫₀^π/⁴ cos² 8θ dθ