{"id":260738,"date":"2025-07-19T16:30:41","date_gmt":"2025-07-19T16:30:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=260738"},"modified":"2025-07-19T16:30:43","modified_gmt":"2025-07-19T16:30:43","slug":"find-the-total-area-of-the-shaded-regions-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/find-the-total-area-of-the-shaded-regions-2\/","title":{"rendered":"Find the total area of the shaded regions."},"content":{"rendered":"\n

Find the total area of the shaded regions. Ay 10 (-2,8) 8 (2,8) y = 2x^2 6 4 2 y = x^4 – 2x^2 -4 -2 2 4 -2 The area is (Simplify your answer.)<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

The correct answer is 128\/15<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

To determine the total area of the shaded regions, we need to calculate the definite integral representing the area between the two given curves: the upper curve y = 2x^2 and the lower curve y = x^4 – 2x^2.<\/p>\n\n\n\n

1. Find the Limits of Integration<\/strong>
First, we find the points where the two curves intersect by setting their equations equal to each other. These intersection points will give us the limits of integration.
2x^2 = x^4 – 2x^2
0 = x^4 – 4x^2
Factor out x^2:
0 = x^2(x^2 – 4)
Factor the difference of squares (x^2 – 4):
0 = x^2(x – 2)(x + 2)
The solutions are x = 0, x = 2, and x = -2. These are the x-values that bound the shaded regions.<\/p>\n\n\n\n

2. Set Up the Integral<\/strong>
The area between two curves is found by integrating the difference between the upper function and the lower function over the given interval. From the graph, y = 2x^2 is the upper curve and y = x^4 – 2x^2 is the lower curve across the entire shaded area, from x = -2 to x = 2.<\/p>\n\n\n\n

The area (A) is given by the integral:
A = \u222b[-2, 2] (upper curve – lower curve) dx
A = \u222b[-2, 2] ( (2x^2) – (x^4 – 2x^2) ) dx
A = \u222b[-2, 2] (4x^2 – x^4) dx<\/p>\n\n\n\n

3. Use Symmetry to Simplify<\/strong>
Notice that the shaded area is symmetric with respect to the y-axis. The function we are integrating, f(x) = 4x^2 – x^4, is an even function because all powers of x are even. We can simplify the calculation by integrating from 0 to 2 and then multiplying the result by 2.
A = 2 * \u222b[0, 2] (4x^2 – x^4) dx<\/p>\n\n\n\n

4. Evaluate the Integral<\/strong>
First, find the antiderivative of the integrand:
\u222b(4x^2 – x^4) dx = 4(x^3\/3) – (x^5\/5) = (4\/3)x^3 – (1\/5)x^5<\/p>\n\n\n\n

Now, apply the Fundamental Theorem of Calculus to evaluate the definite integral from 0 to 2:
A = 2 * [ (4\/3)x^3 – (1\/5)x^5 ] from 0 to 2
A = 2 * ( [ (4\/3)(2)^3 – (1\/5)(2)^5 ] – [ (4\/3)(0)^3 – (1\/5)(0)^5 ] )
A = 2 * ( [ (4\/3)(8) – (1\/5)(32) ] – 0 )
A = 2 * (32\/3 – 32\/5)<\/p>\n\n\n\n

To subtract the fractions, find a common denominator (15):
A = 2 * ( (32 * 5)\/(15) – (32 * 3)\/(15) )
A = 2 * ( (160 – 96)\/15 )
A = 2 * (64\/15)
A = 128\/15<\/p>\n\n\n\n

The total area of the shaded regions is 128\/15.<\/p>\n\n\n\n

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

Find the total area of the shaded regions. Ay 10 (-2,8) 8 (2,8) y = 2x^2 6 4 2 y = x^4 – 2x^2 -4 -2 2 4 -2 The area is (Simplify your answer.) The Correct Answer and Explanation is: The correct answer is 128\/15. Explanation: To determine the total area of the shaded regions, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-260738","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/260738","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=260738"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/260738\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=260738"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=260738"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=260738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}