8 FURTHER APPLICATIONS OF INTEGRATION FURTHER APPLICATIONS OF
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8 FURTHER APPLICATIONS OF INTEGRATION
FURTHER APPLICATIONS OF INTEGRATION 8. 2 Area of a Surface of Revolution In this section, we will learn about: The area of a surface curved out by a revolving arc.
SURFACE OF REVOLUTION A surface of revolution is formed when a curve is rotated about a line. § Such a surface is the lateral boundary of a solid of revolution of the type discussed in Sections 6. 2 and 6. 3
AREA OF A SURFACE OF REVOLUTION We want to define the area of a surface of revolution in such a way that it corresponds to our intuition. § If the surface area is A, we can imagine that painting the surface would require the same amount of paint as does a flat region with area A.
AREA OF A SURFACE OF REVOLUTION Let’s start with some simple surfaces.
CIRCULAR CYLINDERS The lateral surface area of a circular cylinder with radius r and height h is taken to be: A = 2πrh § We can imagine cutting the cylinder and unrolling it to obtain a rectangle with dimensions of 2πrh and h.
CIRCULAR CONES We can take a circular cone with base radius r and slant height l, cut it along the dashed line as shown, and flatten it to form a sector of a circle with radius and central angle θ = 2πr/l.
CIRCULAR CONES We know that, in general, the area of a sector of a circle with radius l and angle θ is ½ l 2 θ.
CIRCULAR CONES So, the area is: § Thus, we define the lateral surface area of a cone to be A = πrl.
AREA OF A SURFACE OF REVOLUTION What about more complicated surfaces of revolution?
AREA OF A SURFACE OF REVOLUTION If we follow the strategy we used with arc length, we can approximate the original curve by a polygon. § When this is rotated about an axis, it creates a simpler surface whose surface area approximates the actual surface area. § By taking a limit, we can determine the exact surface area.
BANDS Then, the approximating surface consists of a number of bands—each formed by rotating a line segment about an axis.
BANDS To find the surface area, each of these bands can be considered a portion of a circular cone.
BANDS Equation 1 The area of the band (or frustum of a cone) with slant height l and upper and lower radii r 1 and r 2 is found by subtracting the areas of two cones:
BANDS From similar triangles, we have: This gives:
BANDS Formula 2 Putting this in Equation 1, we get or where r = ½(r 1 + r 2) is the average radius of the band.
AREA OF A SURFACE OF REVOLUTION Now, we apply this formula to our strategy.
SURFACE AREA Consider the surface shown here. § It is obtained by rotating the curve y = f(x), a ≤ x ≤ b, about the x-axis, where f is positive and has a continuous derivative.
SURFACE AREA To define its surface area, we divide the interval [a, b] into n subintervals with endpoints x 0, x 1, . . . , xn and equal width Δx, as we did in determining arc length.
SURFACE AREA If yi = f(xi), then the point Pi(xi, yi) lies on the curve. § The part of the surface between xi– 1 and xi is approximated by taking the line segment Pi– 1 Pi and rotating it about the x-axis.
SURFACE AREA The result is a band with slant height l = | Pi– 1 Pi | and average radius r = ½(yi– 1 + yi). § So, by Formula 2, its surface area is:
SURFACE AREA As in the proof of Theorem 2 in Section 8. 1, we have where xi* is some number in [xi– 1, xi].
SURFACE AREA When Δx is small, we have yi = f(xi) ≈ f(xi*) and yi– 1 = f(xi– 1) ≈ f(xi*), since f is continuous. Therefore,
SURFACE AREA Formula 3 Thus, an approximation to what we think of as the area of the complete surface of revolution is:
SURFACE AREA The approximation appears to become better as n → ∞.
SURFACE AREA Then, recognizing Formula 3 as a Riemann sum for the function we have:
SURFACE AREA—DEFINITION Formula 4 Thus, in the case where f is positive and has a continuous derivative, we define the surface area of the surface obtained by rotating the curve y = f(x), a ≤ x≤ b, about the x-axis as:
SURFACE AREA Formula 5 With the Leibniz notation for derivatives, this formula becomes:
SURFACE AREA Formula 6 If the curve is described as x = g(y), c ≤ y ≤ d, then the formula for surface area becomes:
SURFACE AREA Formula 7 Then, both Formulas 5 and 6 can be summarized symbolically—using the notation for arc length given in Section 8. 1—as:
Formula 8 SURFACE AREA For rotation about the y-axis, the formula becomes: § Here, as before, we can use either or
SURFACE AREA—FORMULAS You can remember these formulas in the following ways.
SURFACE AREA—FORMULAS Think of 2πy as the circumference of a circle traced out by the point (x, y) on the curve as it is rotated about the x-axis.
SURFACE AREA—FORMULAS Think of 2πx s the circumference of a circle traced out by the point (x, y) on the curve as it is rotated about the y-axis.
Example 1 SURFACE AREA The curve , – 1 ≤ x ≤ 1, is an arc of the circle x 2 + y 2 = 4. Find the area of the surface obtained by rotating this arc about the x-axis. § The surface is a portion of a sphere of radius 2.
SURFACE AREA We have: Example 1
SURFACE AREA Example 1 So, by Formula 5, the surface area is:
SURFACE AREA Example 2 The arc of the parabola y = x 2 from (1, 1) to (2, 4) is rotated about the y-axis. Find the area of the resulting surface.
SURFACE AREA Using y = x 2 and dy/dx = 2 x, from Formula 8, we have: E. g. 2—Solution 1
SURFACE AREA E. g. 2—Solution 1 Substituting u = 1 + 4 x 2, we have du = 8 x dx. Remembering to change the limits of integration, we have:
SURFACE AREA E. g. 2—Solution 2 Using x= and dx/dy = , we have the following solution.
SURFACE AREA E. g. 2—Solution 2
SURFACE AREA Example 3 Find the area of the surface generated by rotating the curve y = ex, 0 ≤ x ≤ 1, about the x-axis.
SURFACE AREA Example 3 Using Formula 5 with y = ex and dy/dx = ex, we have:
SURFACE AREA Example 3
SURFACE AREA Example 3 Since tan α = e , we have: sec 2α = 1 + tan α = 1 + e 2 Thus,
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