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'Find the following definite integrals.'

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'Proof of the Properties of Definite Integrals (A), (B), (C)'

'(2) \ 2 a x-a^{2}=2 b x-b^{2} \ Let us denote that \\( 2(a-b) x=a^{2}-b^{2} \\) Therefore, the \ x \ coordinate of the intersection of \ \\ell_{1} \ and \ \\ell_{2} \ is\n\n\ x=\\frac{a+b}{2} \\n\nTherefore, the area \\( S(a) \\) enclosed by \ \\ell_{1}, \\ell_{2} \ and the parabola C is, from the right figure\n\n\\[\n\egin{aligned}\nS(a)= & \\int_{b}^{\\frac{a+b}{2}}\\left\\{x^{2}-\\left(2 b x-b^{2}\\right)\\right\\} d x \\\\\n& +\\int_{\\frac{a+b}{2}}^{a}\\left\\{x^{2}-\\left(2 a x-a^{2}\\right)\\right\\} d x \\\\\n= & \\int_{b}^{\\frac{a+b}{2}}(x-b)^{2} d x+\\int_{\\frac{a+b}{2}}^{a}(x-a)^{2} d x \\\\\n= & {\\left[\\frac{(x-b)^{3}}{3}\\right]_{b}^{\\frac{a+b}{2}}+\\left[\\frac{(x-a)^{3}}{3}\\right]_{\\frac{a+b}{2}}^{a} } \\\\\n= & \\frac{1}{3}\\left(\\frac{a-b}{2}\\right)^{3}-\\frac{1}{3}\\left(\\frac{b-a}{2}\\right)^{3} \\\\\n= & \\frac{1}{12}(a-b)^{3}=\\frac{1}{12}\\left(a+\\frac{1}{4 a}\\right)^{3}\n\\end{aligned} \\]'

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'Find the following definite integrals.'

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Verify the following equation for subsets $A, B$ of the universal set $U$ using a diagram: \( \overline{(ar{A} \cap B)}=A \cup ar{B} \)