{"id":35191,"date":"2025-06-24T08:02:26","date_gmt":"2025-06-24T08:02:26","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=35191"},"modified":"2025-06-24T08:02:28","modified_gmt":"2025-06-24T08:02:28","slug":"use-the-property-to-estimate-the-best-possible-bounds-of-the-integral","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/use-the-property-to-estimate-the-best-possible-bounds-of-the-integral\/","title":{"rendered":"Use the property to estimate the best possible bounds of the integral"},"content":{"rendered":"\n<p>Use the property to estimate the best possible bounds of the integral: fI 6 sin4(x + Y) dA, T is the triangle enclosed by the lines y = 0, Y = 4X, and x = 1_ &lt; fI 6 sin4(x + Y) dA &lt;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"944\" height=\"302\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-275.png\" alt=\"\" class=\"wp-image-35192\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-275.png 944w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-275-300x96.png 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-275-768x246.png 768w\" sizes=\"auto, (max-width: 944px) 100vw, 944px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To estimate the best possible bounds for the double integral $$\\iint_T 6 \\sin^4(x + y)\\, dA,$$ where TT is the triangular region enclosed by the lines y=0y = 0, y=4xy = 4x, and x=1x = 1, we apply a standard bounding technique.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Analyze the Region<\/h3>\n\n\n\n<p>The triangle TT has vertices at:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(0,0)(0, 0) from the intersection of y=0y = 0 and y=4xy = 4x,<\/li>\n\n\n\n<li>(1,0)(1, 0) from x=1x = 1 and y=0y = 0,<\/li>\n\n\n\n<li>(1,4)(1, 4) from x=1x = 1 and y=4xy = 4x.<\/li>\n<\/ul>\n\n\n\n<p>This forms a right triangle with base along the x-axis from x=0x = 0 to x=1x = 1 and height rising up to y=4y = 4 at x=1x = 1.<\/p>\n\n\n\n<p>The area of this triangle is:<\/p>\n\n\n\n<p>Area=12\u00d7base\u00d7height=12\u00d71\u00d74=2.\\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times 1 \\times 4 = 2.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Understand the Integrand<\/h3>\n\n\n\n<p>The integrand is 6sin\u20614(x+y)6 \\sin^4(x + y). Since sine values always lie in [\u22121,1][-1, 1], raising sine to the fourth power produces values in [0,1][0, 1]. Multiplying by 6 scales this to the range [0,6][0, 6]. Therefore, across the entire region TT, the function 6sin\u20614(x+y)6 \\sin^4(x + y) takes values between 0 and 6.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Apply the Bounding Property<\/h3>\n\n\n\n<p>We now use the property that if m\u2264f(x,y)\u2264Mm \\leq f(x, y) \\leq M for all (x,y)\u2208T(x, y) \\in T, then:<\/p>\n\n\n\n<p>m\u22c5Area(T)\u2264\u222cTf(x,y)\u2009dA\u2264M\u22c5Area(T).m \\cdot \\text{Area}(T) \\leq \\iint_T f(x, y)\\, dA \\leq M \\cdot \\text{Area}(T).<\/p>\n\n\n\n<p>Substituting m=0m = 0, M=6M = 6, and Area=2\\text{Area} = 2:<\/p>\n\n\n\n<p>0\u2264\u222cT6sin\u20614(x+y)\u2009dA\u226412.0 \\leq \\iint_T 6 \\sin^4(x + y)\\, dA \\leq 12.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer<\/h3>\n\n\n\n<p>0&lt;\u222cT6sin\u20614(x+y)\u2009dA&lt;12\\boxed{0 &lt; \\iint_T 6 \\sin^4(x + y)\\, dA &lt; 12}<\/p>\n\n\n\n<p>This result gives the best possible bounds without evaluating the integral directly.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-638.jpeg\" alt=\"\" class=\"wp-image-35193\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-638.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-638-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-638-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Use the property to estimate the best possible bounds of the integral: fI 6 sin4(x + Y) dA, T is the triangle enclosed by the lines y = 0, Y = 4X, and x = 1_ &lt; fI 6 sin4(x + Y) dA &lt; The Correct Answer and Explanation is: To estimate the best possible [&hellip;]<\/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-35191","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35191","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=35191"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35191\/revisions"}],"predecessor-version":[{"id":35194,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/35191\/revisions\/35194"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=35191"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=35191"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=35191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}