{"id":36868,"date":"2025-06-25T09:58:26","date_gmt":"2025-06-25T09:58:26","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=36868"},"modified":"2025-06-25T09:58:27","modified_gmt":"2025-06-25T09:58:27","slug":"find-the-exact-area-of-the-shaded-region-where-fx1-2xex","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-area-of-the-shaded-region-where-fx1-2xex\/","title":{"rendered":"Find the exact area of the shaded region, where\u00a0f(x)=1\/2xex\u00a0"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To find the exact area of the shaded region between the curves f(x)=12xexf(x) = \\frac{1}{2}xe^x and g(x)=x2ln\u2061xg(x) = x^2 \\ln x over the interval x=1x = 1 to x=2x = 2, we evaluate the definite integral of their difference:<\/p>\n\n\n\n

Area=\u222b12(12xex\u2212x2ln\u2061x)dx\\text{Area} = \\int_{1}^{2} \\left( \\frac{1}{2}xe^x – x^2 \\ln x \\right) dx<\/p>\n\n\n\n

We start with the integral of 12xex\\frac{1}{2}xe^x. Using integration by parts:<\/p>\n\n\n\n

Let u=xu = x, dv=exdxdv = e^x dx, so du=dxdu = dx and v=exv = e^x. Then,<\/p>\n\n\n\n

\u222bxexdx=xex\u2212\u222bexdx=xex\u2212ex\\int xe^x dx = xe^x – \\int e^x dx = xe^x – e^x<\/p>\n\n\n\n

Thus,<\/p>\n\n\n\n

\u222b12xexdx=12(xex\u2212ex)=12xex\u221212ex\\int \\frac{1}{2}xe^x dx = \\frac{1}{2}(xe^x – e^x) = \\frac{1}{2}xe^x – \\frac{1}{2}e^x<\/p>\n\n\n\n

Evaluating from 1 to 2:<\/p>\n\n\n\n

[12xex\u221212ex]12=(e2\u221212e2)\u2212(12e\u221212e)=12e2\\left[\\frac{1}{2}xe^x – \\frac{1}{2}e^x\\right]_1^2 = \\left(e^2 – \\frac{1}{2}e^2\\right) – \\left(\\frac{1}{2}e – \\frac{1}{2}e\\right) = \\frac{1}{2}e^2<\/p>\n\n\n\n

Now compute the integral of x2ln\u2061xx^2 \\ln x. Use integration by parts again:<\/p>\n\n\n\n

Let u=ln\u2061xu = \\ln x, dv=x2dxdv = x^2 dx, then du=1xdxdu = \\frac{1}{x} dx, v=x33v = \\frac{x^3}{3}<\/p>\n\n\n\n

\u222bx2ln\u2061xdx=x33ln\u2061x\u2212\u222bx33\u22c51xdx=x33ln\u2061x\u221213\u222bx2dx=x33ln\u2061x\u2212x39\\int x^2 \\ln x dx = \\frac{x^3}{3} \\ln x – \\int \\frac{x^3}{3} \\cdot \\frac{1}{x} dx = \\frac{x^3}{3} \\ln x – \\frac{1}{3} \\int x^2 dx = \\frac{x^3}{3} \\ln x – \\frac{x^3}{9}<\/p>\n\n\n\n

Evaluate from 1 to 2:<\/p>\n\n\n\n

[x33ln\u2061x\u2212x39]12=(83ln\u20612\u221289)\u2212(0\u221219)=83ln\u20612\u221279\\left[\\frac{x^3}{3} \\ln x – \\frac{x^3}{9}\\right]_1^2 = \\left(\\frac{8}{3}\\ln 2 – \\frac{8}{9}\\right) – (0 – \\frac{1}{9}) = \\frac{8}{3}\\ln 2 – \\frac{7}{9}<\/p>\n\n\n\n

So the exact area is:<\/p>\n\n\n\n

12e2\u2212(83ln\u20612\u221279)=12e2\u221283ln\u20612+79\\frac{1}{2}e^2 – \\left(\\frac{8}{3} \\ln 2 – \\frac{7}{9}\\right) = \\frac{1}{2}e^2 – \\frac{8}{3}\\ln 2 + \\frac{7}{9}<\/p>\n\n\n\n

Correct Answer:<\/strong><\/p>\n\n\n\n

79+12e2\u221283ln\u20612\\boxed{\\frac{7}{9} + \\frac{1}{2}e^2 – \\frac{8}{3} \\ln 2}<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

The Correct Answer and Explanation is: To find the exact area of the shaded region between the curves f(x)=12xexf(x) = \\frac{1}{2}xe^x and g(x)=x2ln\u2061xg(x) = x^2 \\ln x over the interval x=1x = 1 to x=2x = 2, we evaluate the definite integral of their difference: Area=\u222b12(12xex\u2212x2ln\u2061x)dx\\text{Area} = \\int_{1}^{2} \\left( \\frac{1}{2}xe^x – x^2 \\ln x \\right) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-36868","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36868","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=36868"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36868\/revisions"}],"predecessor-version":[{"id":36881,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36868\/revisions\/36881"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=36868"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=36868"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=36868"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}