Question

Q5: Consider the region bounded by the curves y = 2 - |x| and y = x^4 a) Sketch the region and find its area y = x^4 ? upper - lower ?_{-1}^{1} -|x| + 2 - x^4 dx = [-|x|x/2 + 2x - 1/5x^5]_{-1}^{1} = (-|1|1/2 + 2 - 1/5) - (-|-1|(-1)/2 - 2 + 1/5) = (-1/2 + 9/5) - (1/2 - 9/5) = 13/10 - (-13/10) = 26/10 = 13/5 B) By integration, find the volume of the solid obtained by rotating the given region about the x-axis

          Q5: Consider the region bounded by the curves y = 2 - |x| and y = x^4

a) Sketch the region and find its area

y = x^4

? upper - lower

?_{-1}^{1} -|x| + 2 - x^4 dx

= [-|x|x/2 + 2x - 1/5x^5]_{-1}^{1}

= (-|1|1/2 + 2 - 1/5) - (-|-1|(-1)/2 - 2 + 1/5)

= (-1/2 + 9/5) - (1/2 - 9/5)

= 13/10 - (-13/10)

= 26/10 = 13/5

B) By integration, find the volume of the solid obtained by rotating the given region about the x-axis

Q5: Consider the region bounded by the curves y = 2 - |x| and y = x^4

a) Sketch the region and find its area

y = x^4

? upper - lower

?-1^1 -|x| + 2 - x^4 dx

= [-|x|x/2 + 2x - 1/5x^5]-1^1

= (-|1|1/2 + 2 - 1/5) - (-|-1|(-1)/2 - 2 + 1/5)

= (-1/2 + 9/5) - (1/2 - 9/5)

= 13/10 - (-13/10)

= 26/10 = 13/5

B) By integration, find the volume of the solid obtained by rotating the given region about the x-axis

Added by Luke R.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/16/2023

Step 1

For $x \geq 0$, we have $y = 2 - x$. Solving the system of equations, we get: $$2 - x = -x^2 + 2x + 9$$ $$x^2 - 3x - 7 = 0$$ By the quadratic formula, we find that $x = \frac{3 + \sqrt{37}}{2}$ is the positive solution. For $x < 0$, we have $y = 2 + x$. Solving Show more…

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Consider the region bounded by the curves y = 2 - |x| and y = x^4. a) Sketch the region and find its area. b) By integration, find the volume of the solid obtained by rotating the given region about the x-axis.