Textbook Question
Determine the area of the shaded region in the following figures.
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9. Graphical Applications of Integrals
Area Between Curves
4:04 minutesProblem 6.2.35a
Textbook Question
For the given regions R₁ and R₂, complete the following steps.
a. Find the area of region R₁.
R₁is the region in the first quadrant bounded by the line x=1 and the curve y=6x(2−x^2)^2; R₂ is the region in the first quadrant bounded the curve y=6x(2−x^2)^2and the line y=6x.
Verified step by step guidance1
Identify the boundaries of region R₁. It is bounded by the line $x=1$, the curve $y=6x(2 - x^2)^2$, and the coordinate axes in the first quadrant. Since it is in the first quadrant, $x$ ranges from 0 to 1.
Set up the integral for the area of region R₁. Because the region is bounded vertically by the curve $y=6x(2 - x^2)^2$ and horizontally by $x=1$, the area can be found by integrating the function $y=6x(2 - x^2)^2$ with respect to $x$ from 0 to 1:
$$\text{Area}_{R_1} = \int_0^1 6x(2 - x^2)^2 \, dx.$$
Before integrating, consider expanding the integrand $6x(2 - x^2)^2$ to simplify the integral. First, expand $(2 - x^2)^2$ using the binomial formula:
$$(2 - x^2)^2 = 4 - 4x^2 + x^4.$$
Multiply this expansion by $6x$ to get the integrand in polynomial form, then integrate term-by-term over the interval $[0,1]$ to find the area of region R₁.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals for Area Calculation
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