Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).
9. Graphical Applications of Integrals
Area Between Curves
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Find the area of the following regions, expressing your results in terms of the positive integer n≥2.
Let Aₙ be the area of the region bounded by f(x)=x^1/n and g(x)=x^n on the interval [0,1], where n is a positive integer. Evaluate lim n→∞ Aₙ and interpret the result. br
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27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Find the area of each of the regions R₁,R₂, and R₃.
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14–25. {Use of Tech} Areas of regions Determine the area of the given region.
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Express the area of the shaded region in Exercise 5 as the sum of two integrals with respect to y. Do not evaluate the integrals.
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Find the area of the region (see figure) in two ways.
a. Using integration with respect to x.
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Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x=b.
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21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = ln x, for 1≤x≤4
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21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = x³/3, for −1≤x≤1
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21–30. {Use of Tech} Arc length by calculator
a. Write and simplify the integral that gives the arc length of the following curves on the given interval.
y = cos 2x, for 0 ≤ x ≤ π
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21–30. {Use of Tech} Arc length by calculator
b. If necessary, use technology to evaluate or approximate the integral.
y = cos 2x, for 0 ≤ x ≤ π
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Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
b. ∫a^b √1+36 cos² 2xdx
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35-38. Area and volume Let R be the region in the first quadrant bounded by the graph of
Find the area of the region R.
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An area function Consider the functions y = x²/a and y = √x/a, where a>0. Find A(a), the area of the region between the curves.
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Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.
a. ∫a^b √1+16x⁴ dx