{"id":36858,"date":"2025-06-25T09:53:34","date_gmt":"2025-06-25T09:53:34","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=36858"},"modified":"2025-06-25T09:53:35","modified_gmt":"2025-06-25T09:53:35","slug":"find-the-exact-area-of-the-shaded-region-in-figure-6-25-betweenand-the-axis","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-exact-area-of-the-shaded-region-in-figure-6-25-betweenand-the-axis\/","title":{"rendered":"Find the exact area of the shaded region in Figure 6.25 betweenand the-axis."},"content":{"rendered":"\n

Find the exact area of the shaded region in Figure 6.25 between
and the
-axis. (FIGURE<\/p>\n\n\n\n

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

To help you correctly, I need to see Figure 6.25<\/strong> or at least know what curve or function is graphed and the interval along the x-axis over which the shaded region lies.<\/p>\n\n\n\n

However, I can still demonstrate<\/strong> the general process for finding the exact area between a curve and the x-axis<\/strong> using an example. Suppose we are given a function:f(x)=x2f(x) = x^2f(x)=x2<\/p>\n\n\n\n

and the shaded region lies between x=1x = 1x=1 and x=3x = 3x=3. The area between this curve and the x-axis is calculated using definite integration<\/strong>:<\/p>\n\n\n\n

\n\n\n\n

Step-by-step Process:<\/h3>\n\n\n\n
    \n
  1. Set up the integral<\/strong>:<\/li>\n<\/ol>\n\n\n\n

    Area=\u222b13×2\u2009dx\\text{Area} = \\int_{1}^{3} x^2 \\, dxArea=\u222b13\u200bx2dx<\/p>\n\n\n\n

      \n
    1. Find the antiderivative<\/strong> of x2x^2×2:<\/li>\n<\/ol>\n\n\n\n

      \u222bx2\u2009dx=x33\\int x^2 \\, dx = \\frac{x^3}{3}\u222bx2dx=3×3\u200b<\/p>\n\n\n\n

        \n
      1. Evaluate the definite integral<\/strong>:<\/li>\n<\/ol>\n\n\n\n

        x33\u222313=333\u2212133=273\u221213=263\\left. \\frac{x^3}{3} \\right|_{1}^{3} = \\frac{3^3}{3} – \\frac{1^3}{3} = \\frac{27}{3} – \\frac{1}{3} = \\frac{26}{3}3×3\u200b\u200b13\u200b=333\u200b\u2212313\u200b=327\u200b\u221231\u200b=326\u200b<\/p>\n\n\n\n

        \n\n\n\n

        Final Answer:<\/h3>\n\n\n\n

        263 square units\\boxed{\\frac{26}{3}} \\text{ square units}326\u200b\u200b square units<\/p>\n\n\n\n

        \n\n\n\n

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

        The area between a curve and the x-axis<\/strong> over a given interval [a,b][a, b][a,b] is found by evaluating the definite integral<\/strong> of the function f(x)f(x)f(x) from aaa to bbb. If the function is always positive<\/strong> over the interval, the integral gives the exact area. If the function dips below the x-axis, you may need to take the absolute value<\/strong> of those portions or split the integral into parts.<\/p>\n\n\n\n

        In our example, x2x^2×2 is always positive or zero, so the definite integral directly gives the total area. You can follow the same steps once you provide the actual figure or function in your question. Please upload or describe Figure 6.25<\/strong>, and I will solve it precisely.<\/p>\n\n\n\n

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

        Find the exact area of the shaded region in Figure 6.25 betweenand the-axis. (FIGURE The Correct Answer and Explanation is: To help you correctly, I need to see Figure 6.25 or at least know what curve or function is graphed and the interval along the x-axis over which the shaded region lies. However, I can […]<\/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-36858","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36858","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=36858"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36858\/revisions"}],"predecessor-version":[{"id":36866,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36858\/revisions\/36866"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=36858"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=36858"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=36858"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}