Answer to Question #242217 in Calculus for Josh

Question #242217


Find the area A(R) of the region R described below.


1. R is the region bounded by y = 2x + 3, the x-axis and the line x=5 .


2. R is the region bounded by y=x^{2}+1 , the lines y=2 and y=5


3. R is the region bounded by x=y(8-y) and the y-axis.


4. R is the region bounded by y = x and y=x^{3} .


5. R is the region bounded by y=2-x^{2} and y=x^{2}-6


1
Expert's answer
2021-09-29T11:48:03-0400

1.

"y\\ =\\ 2x\\ +\\ 3\\\\\nx=\\frac{1}{2}y-\\frac{3}{2}\\\\\nx\\ intercept\\ is-\\frac{3}{2}\\\\\n\\int_{-\\frac{3}{2}}^{5}{2x+3}\\ dx\\\\\nx^2+3x\\ |_{-1.5}^5\\\\\n40\\ -\\ -2.25\\ =\\ 42.25\\ square\\ units"


2.

"y=x^2+1"

"x=\\left(y-1\\right)^{0.5}\n\\\\area\\ of\\ R=2\\ \\int_{2}^{5}{\\left(y-1\\right)^{0.5}dy}\\\\\n2\\left(\\frac{\\left(y-1\\right)^{1.5}}{1.5}\\right)\\ |_2^5\\\\\n2\\left(\\frac{4^{1.5}-1}{1.5}\\right)=9.333\\ square\\ units"


3.

"x=y\\left(8-y\\right)\\\\\nat\\ the\\ y\\ axis,\\ x=0\\\\\ny\\left(8-y\\right)=0\\\\\ny=0\\ or\\ y=8\\\\"

"\\int_{0}^{8}{8y-y^2}\\ dy\n\\\\4y^2-\\frac{y^3}{3}\\ \n\\\\4\\left(64\\right)-\\frac{8^3}{3}=85.3333"


4.

"\\\\y=x\n\\\\y=x^3\n\\\\x=x^3\n\\\\x\\left(1-x^2\\right)=0\n\\\\x=0,\\ x=1,\\ x=\\ -1"


"Area\\ of\\ R=\\int_{-1}^{0}{x^3-x}\\ dx+\\ \\int_{0}^{1}{x-x^3dx}\\\\\n\\frac{x^4}{4}-\\frac{x^2}{2}|_{-1}^0+\\frac{x^2}{2}-\\frac{x^4}{4}\\ |_0^1\\ \n\n\\\\-\\frac{1}{4}+\\frac{1}{2}+\\frac{1}{2}-\\frac{1}{4}=0.5\\ square\\ units"


5.

"\\\\y=2-x^2\n\\\\y=x^2-6\n\\\\2-x^2=x^2-6\n\\\\2x^2-8=0\n\\\\x=\\ 2,\\ x=\\ -2"


"Area\\ of\\ R=\\ \\int_{-2}^{2}{2-x^2-\\left(x^2-6\\right)}dx\\\\\n\\int_{-2}^{2}{8-2x^2}\\ dx\\\\\n8x-\\frac{2}{3}x^3|_{-2}^2\\\\\n10\\frac{2}{3}--10\\frac{2}{3}=21\\frac{1}{3}\\ square\\ units"


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