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Niaboc67
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. Homework Statement
Q1. Let f(x) be any continuous function that satisfies: $$-2x≤xg(x)≤2x$$ for $$0≤x≤1$$ Find the upper and lower bounds for:
$$\int_{0}^{1} \sqrt{1+g(x)+x^2}dx$$
Q2. Let h(x) be any continuous function that satisfies: $$-4≤h(x)≤x^2-4$$ for $$0≤x≤1$$ Find the upper and lower bounds for:
$$\int_{0}^{1}x^3h(x)dx$$
Soltn for Q1 for lower bounds: $$\int_{0}^{1} \sqrt{1+g(x)+x^2}dx \\
= \int_{0}^{1} \sqrt{1-2x+x^2}dx \\
= \int_{0}^{1} \sqrt{(x-1)^2}dx \\
L_{f}(P) = \int_{0}^{1} (x-1)dx$$
Soltn for Q2 for upper bounds
$$\int_{0}^{1} \sqrt{1+2x+x^2}dx \\
= \int_{0}^{1} \sqrt{(x+1)^2}dx \\
U_{f}(P) = \int_{0}^{1} (x+1)dx$$Soltn for Q2 for lower bound: $$\int_{0}^{1}x^3h(x)dx \\
= \int_{0}^{1}x^3(-4)dx \\
L_{f}(P) = \int_{0}^{1}-4x^3dx$$
Soltn for Q2 for upper bound...
$$\int_{0}^{1}x^3(x^2-4)dx\\
U_{f}(P) = \int_{0}^{1}x^6-4x^3dx$$Thank you
Q1. Let f(x) be any continuous function that satisfies: $$-2x≤xg(x)≤2x$$ for $$0≤x≤1$$ Find the upper and lower bounds for:
$$\int_{0}^{1} \sqrt{1+g(x)+x^2}dx$$
Q2. Let h(x) be any continuous function that satisfies: $$-4≤h(x)≤x^2-4$$ for $$0≤x≤1$$ Find the upper and lower bounds for:
$$\int_{0}^{1}x^3h(x)dx$$
The Attempt at a Solution
Soltn for Q1 for lower bounds: $$\int_{0}^{1} \sqrt{1+g(x)+x^2}dx \\
= \int_{0}^{1} \sqrt{1-2x+x^2}dx \\
= \int_{0}^{1} \sqrt{(x-1)^2}dx \\
L_{f}(P) = \int_{0}^{1} (x-1)dx$$
Soltn for Q2 for upper bounds
$$\int_{0}^{1} \sqrt{1+2x+x^2}dx \\
= \int_{0}^{1} \sqrt{(x+1)^2}dx \\
U_{f}(P) = \int_{0}^{1} (x+1)dx$$Soltn for Q2 for lower bound: $$\int_{0}^{1}x^3h(x)dx \\
= \int_{0}^{1}x^3(-4)dx \\
L_{f}(P) = \int_{0}^{1}-4x^3dx$$
Soltn for Q2 for upper bound...
$$\int_{0}^{1}x^3(x^2-4)dx\\
U_{f}(P) = \int_{0}^{1}x^6-4x^3dx$$Thank you
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