{"id":45373,"date":"2025-07-01T06:34:47","date_gmt":"2025-07-01T06:34:47","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=45373"},"modified":"2025-07-01T06:34:49","modified_gmt":"2025-07-01T06:34:49","slug":"consider-the-integral-given-by","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/consider-the-integral-given-by\/","title":{"rendered":"Consider the integral given by"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

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

Here is the step-by-step solution to the problem.<\/p>\n\n\n\n

Part (a): Determine \u0394x and x\u1d62.<\/h3>\n\n\n\n

To find \u0394x and x\u1d62, we first identify the components of the given integral, \u222b\u00b9\u2082 (2x\u00b3) dx, in the context of the definition of a definite integral.<\/p>\n\n\n\n

    \n
  • The function is f(x) = 2x\u00b3.<\/li>\n\n\n\n
  • The lower limit of integration is a = 1.<\/li>\n\n\n\n
  • The upper limit of integration is b = 2.<\/li>\n<\/ul>\n\n\n\n

    1. Calculate \u0394x:<\/strong>
    The width of each subinterval, \u0394x, is given by the formula:
    \u0394x = (b – a) \/ n
    Substituting the values of a and b:
    \u0394x = (2 – 1) \/ n = 1\/n<\/p>\n\n\n\n

    2. Determine x\u1d62:<\/strong>
    The problem specifies using right-hand endpoints. The formula for the right-hand endpoint of the i-th subinterval is:
    x\u1d62 = a + i\u0394x
    Substituting the values for a and \u0394x:
    x\u1d62 = 1 + i(1\/n) = 1 + i\/n<\/p>\n\n\n\n

    Answers for Part (a):<\/strong>
    \u0394x =<\/strong> 1\/n
    x\u1d62 =<\/strong> 1 + i\/n<\/p>\n\n\n\n

    \n\n\n\n

    Part (b): Using the definition mentioned above, evaluate the integral.<\/h3>\n\n\n\n

    We use the definition of the definite integral as the limit of a Riemann sum:
    \u222b\u2090\u1d47 f(x) dx = lim(n\u2192\u221e) \u03a3\u1d62\u208c\u2081\u207f f(x\u1d62)\u0394x<\/p>\n\n\n\n

    1. Set up the sum:<\/strong>
    Substitute f(x\u1d62) and \u0394x into the formula:
    lim(n\u2192\u221e) \u03a3\u1d62\u208c\u2081\u207f [2(1 + i\/n)\u00b3] * (1\/n)<\/p>\n\n\n\n

    2. Expand and simplify the expression:<\/strong>
    First, expand the cubic term (1 + i\/n)\u00b3:
    (1 + i\/n)\u00b3 = 1\u00b3 + 3(1)\u00b2(i\/n) + 3(1)(i\/n)\u00b2 + (i\/n)\u00b3 = 1 + 3i\/n + 3i\u00b2\/n\u00b2 + i\u00b3\/n\u00b3
    Now, substitute this back into the sum:
    lim(n\u2192\u221e) \u03a3\u1d62\u208c\u2081\u207f [2(1 + 3i\/n + 3i\u00b2\/n\u00b2 + i\u00b3\/n\u00b3)] * (1\/n)
    Distribute the 2 and the 1\/n:
    lim(n\u2192\u221e) \u03a3\u1d62\u208c\u2081\u207f (2\/n + 6i\/n\u00b2 + 6i\u00b2\/n\u00b3 + 2i\u00b3\/n\u2074)<\/p>\n\n\n\n

    3. Apply summation properties and formulas:<\/strong>
    Separate the summation term by term and factor out constants:
    lim(n\u2192\u221e) [ (2\/n)\u03a3\u1d62\u208c\u2081\u207f 1 + (6\/n\u00b2)\u03a3\u1d62\u208c\u2081\u207f i + (6\/n\u00b3)\u03a3\u1d62\u208c\u2081\u207f i\u00b2 + (2\/n\u2074)\u03a3\u1d62\u208c\u2081\u207f i\u00b3 ]
    Use the standard summation formulas:<\/p>\n\n\n\n

      \n
    • \u03a3\u1d62\u208c\u2081\u207f 1 = n<\/li>\n\n\n\n
    • \u03a3\u1d62\u208c\u2081\u207f i = n(n+1)\/2<\/li>\n\n\n\n
    • \u03a3\u1d62\u208c\u2081\u207f i\u00b2 = n(n+1)(2n+1)\/6<\/li>\n\n\n\n
    • \u03a3\u1d62\u208c\u2081\u207f i\u00b3 = [n(n+1)\/2]\u00b2 = n\u00b2(n+1)\u00b2\/4<\/li>\n<\/ul>\n\n\n\n

      Substitute these formulas into the expression:
      lim(n\u2192\u221e) [ (2\/n)(n) + (6\/n\u00b2)(n(n+1)\/2) + (6\/n\u00b3)(n(n+1)(2n+1)\/6) + (2\/n\u2074)(n\u00b2(n+1)\u00b2\/4) ]<\/p>\n\n\n\n

      4. Simplify and evaluate the limit:<\/strong>
      Simplify each term:
      lim(n\u2192\u221e) [ 2 + 3(n+1)\/n + (n+1)(2n+1)\/n\u00b2 + (n+1)\u00b2\/(2n\u00b2) ]
      lim(n\u2192\u221e) [ 2 + 3(1 + 1\/n) + (2n\u00b2+3n+1)\/n\u00b2 + (n\u00b2+2n+1)\/(2n\u00b2) ]
      lim(n\u2192\u221e) [ 2 + 3(1 + 1\/n) + (2 + 3\/n + 1\/n\u00b2) + (1\/2 + 1\/n + 1\/(2n\u00b2)) ]
      As n approaches infinity, any term with n in the denominator (like 1\/n or 1\/n\u00b2) approaches 0.
      So, the limit becomes:
      2 + 3(1 + 0) + (2 + 0 + 0) + (1\/2 + 0 + 0) = 2 + 3 + 2 + 0.5 = 7.5<\/p>\n\n\n\n

      Answer for Part (b):<\/strong>
      Value of integral:<\/strong> 7.5<\/p>\n\n\n\n

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

      The Correct Answer and Explanation is: Here is the step-by-step solution to the problem. Part (a): Determine \u0394x and x\u1d62. To find \u0394x and x\u1d62, we first identify the components of the given integral, \u222b\u00b9\u2082 (2x\u00b3) dx, in the context of the definition of a definite integral. 1. Calculate \u0394x:The width of each subinterval, \u0394x, […]<\/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-45373","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45373","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=45373"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45373\/revisions"}],"predecessor-version":[{"id":45376,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/45373\/revisions\/45376"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=45373"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=45373"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=45373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}