Area Between Curves: Find the area of the region bounded by the curves y = f (x) and y = g(x). Area Between Curves: Find the area of the region bounded by the curves y = f (x) and y = g(x). Step 1: Draw a rough sketch the two curves and shade the bounded region between them with vertical lines. Identity the top and bottom function of x. Area Between Curves: Find the area of the region bounded by the curves y = f (x) and y = g(x). Step 1: Draw a rough sketch the two curves and shade the bounded region between them with vertical lines. Identity the top and bottom function of x. Step 2: Find the x-coordinates of the points of intersection. Add these to the sketch. If necessary make a new sketch. Area Between Curves: Find the area of the region bounded by the curves y = f (x) and y = g(x). Step 1: Draw a rough sketch the two curves and shade the bounded region between them with vertical lines. Identity the top and bottom function of x. Step 2: Find the x-coordinates of the points of intersection. Add these to the sketch. If necessary make a new sketch. Step 3: On the sketch, darkly shade a typical rectangle between x = xi and x = xi + ∆x. Write down its area, ∆Ai. Area Between Curves: Find the area of the region bounded by the curves y = f (x) and y = g(x). Step 1: Draw a rough sketch the two curves and shade the bounded region between them with vertical lines. Identity the top and bottom function of x. Step 2: Find the x-coordinates of the points of intersection. Add these to the sketch. If necessary make a new sketch. Step 3: On the sketch, darkly shade a typical rectangle between x = xi and x = xi + ∆x. Write down its area, ∆Ai. Step 4: Write the area A as a deﬁnite integral, simplify using symmetry if possible, and evaluate. Area Between Curves: Find the area of the region bounded by the curves y = x2 and y = 2x2 − 1. Step 1: Draw a rough sketch the two curves and shade the bounded region between them with vertical lines. Identity the top and bottom function of x. Step 2: Find the x-coordinates of the points of intersection. Add these to the sketch. If necessary make a new sketch. Step 3: On the sketch, darkly shade a typical rectangle between x = xi and x = xi + ∆x. Write down its area, ∆Ai. Step 4: Write the area A as a deﬁnite integral, simplify using symmetry if possible, and evaluate. Area Between Curves: Find the area of the region bounded by the curves y = x2 and y = 2x2 − 1. Step 1: Draw a rough sketch the two curves and shade the bounded region between them with vertical lines. Identity the top and bottom function of x. Step 2: Find the x-coordinates of the points of intersection. Add these to the sketch. If necessary make a new sketch. Ans: x = ±1. Step 3: On the sketch, darkly shade a typical rectangle between x = xi and x = xi + ∆x. Write down its area, ∆Ai. Ans: ∆Ai = (1 − x2i )∆x Step 4: Write the area A as a deﬁnite integral, simplify using symmetry if ∫ 1 ∫ 1 possible, and evaluate. Ans: A = −1(1 − x2)dx = 2 0 (1 − x2)dx = 43

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