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Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions

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Objectives To develop a formula for finding the area under a curve.

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Area Under the Curve How do we find areas under a curve, but above the x-axis?

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Area Under the Curve As the number of rectangles used to approximate the area of the region increases, the approximation becomes more accurate.

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Area Under the Curve It is possible to find the exact area by letting the width of each rectangle approach zero. Doing this generates an infinite number of rectangles.

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Area Under the Curve Find the area of the shaded region.

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Area Under the Curve = um (height) (base) Area sum of the areas of the rectangles = = a b The formula looks like an integral.

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Area Under the Curve = a b The formula looks like an integral. Is the area really given by the antiderivative? Yes! The definite integral of f from a to b is the limit of the Riemann sum as the lengths of the subintervals approach zero.

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Area Under the Curve A function and the equation for the area between its graph and the x-axis are related to each other by the antiderivative. = a b The formula looks like an integral.

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Two Questions of Calculus Q 1 : How do you find instantaneous velocity? A: Use the derivative. Q 2 : How do you find the area of exotic shapes? A: Use the antiderivative.

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Area Under the Curve How do we calculate areas under a curve, but above the x-axis?

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The Fundamental Theorem of Calculus

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Find the area between the graph of f and the x-axis on the interval [0, 3].

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The Fundamental Theorem of Calculus Find the area between the graph of f and the x-axis on the interval [0, 3]. The bar tells you to evaluate the expression at 3 and subtract the value of the expression at 0.

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The Fundamental Theorem of Calculus

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Find the area between the graph of f and the x-axis on the interval [0, 1].

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The Fundamental Theorem of Calculus Substitute into the integral. Always express your answer in terms of the original variable.

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The Fundamental Theorem of Calculus So this would represent the area between the curve y = x √(x 2 + 1) and the x-axis from x = 0 to 1

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Conclusion As the number of rectangles used to approximate the area of the region increases, the approximation becomes more accurate. It is possible to find the exact area by letting the width of each rectangle approach zero. Doing this generates an infinite number of rectangles. A function and the equation for the area between its graph and the x-axis are related to each other by the antiderivative. The Fundamental Theorem of Calculus enables us to evaluate definite integrals. This empowers us to find the area between a curve and the x-axis.

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