Ch. 8 – Applications of Definite Integrals

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Presentation transcript:

Ch. 8 – Applications of Definite Integrals 8.2 – Areas in the Plane

Ex: Find the area of the region enclosed by the curves of y = x2 and y = x + 2. It’s area, so set up an integral! Graph the function to assist you…make sure to subtract the bottom curve from the top curve! Given two continuous curves f(x) and g(x) over [a, b], and f(x) ≥ g(x) throughout [a, b], then the area between f and g is:

Introduce integrating with respect to y here (Day 2) Area = 7.5 units2

Ex: Find the area of the region enclosed by the curves of y = cosx and y = sin2x – cosx over [-π/2, π/2]. Set up an integral…remember, top curve minus bottom curve! Graph the function to assist you! Only integrate subtracted functions for area entirely above one curve and entirely below another curve!

Side thought: could the area ever be negative? Ex: Find the area of the region enclosed by the curves of y = 2cosx and y = x2 – 1 using your calculator. Graph, then set up an integral! Find the intersection points using your calculator. The x-value will automatically be stored as X, so store it as another letter so you won’t lose it. Evaluate using fnInt! Answer ≈ 4.995 Side thought: could the area ever be negative?

NOT ALL AREAS WILL BE REGIONS BETWEEN 2 FUNCTIONS! Ex: Use your calculator to find the area of the region enclosed by the curves of x = y2 – 2 and y = x3 . Whenever one equation is not a function, solve for x and switch x and y first. Now graph and integrate. Use your calculator.

Ex: Find the area of the region enclosed by the curves of x - y2 = 0 and x + 2y2 = 3 without a calculator. Solve for x and switch x and y… Find the limits of integration by substitution (set one function equal to the other) Set up the integral and solve! (Which function is on top? Will it matter?) Use Symmetry!