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Calculate the iterated integral. int_(0)^(1)int_(0)^(1)5sqrt(s+t)dsdt

Calculate the iterated integral.\newline∫01∫015s+tdsdt \int_{0}^{1} \int_{0}^{1} 5 \sqrt{s+t} d s d t

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Full solution

Q. Calculate the iterated integral.\newline∫01∫015s+tdsdt \int_{0}^{1} \int_{0}^{1} 5 \sqrt{s+t} d s d t
  1. Given iterated integral: We are given the iterated integral: ∫01∫015s+t ds dt\int_{0}^{1}\int_{0}^{1}5\sqrt{s+t}\,ds\,dt We will first integrate with respect to ss, keeping tt constant.
  2. Integrating with respect to ss: The inner integral is: ∫015s+t ds\int_{0}^{1} 5\sqrt{s+t}\,ds To integrate this, we treat tt as a constant and integrate the function 5s+t5\sqrt{s+t} with respect to ss.
  3. Evaluating antiderivative: The antiderivative of 5s+t5\sqrt{s+t} with respect to ss is:\newline(52)⋅(23)⋅(s+t)32=103⋅(s+t)32(\frac{5}{2})\cdot(\frac{2}{3})\cdot(s+t)^{\frac{3}{2}} = \frac{10}{3}\cdot(s+t)^{\frac{3}{2}}\newlineWe will now evaluate this from s=0s=0 to s=1s=1.
  4. Integrating with respect to t: Plugging in the limits of integration, we get:\newline(103)⋅(1+t)32(\frac{10}{3})\cdot(1+t)^{\frac{3}{2}} - (103)⋅t32(\frac{10}{3})\cdot t^{\frac{3}{2}}\newlineThis simplifies to:\newline(103)⋅((1+t)32−t32)(\frac{10}{3})\cdot\left((1+t)^{\frac{3}{2}} - t^{\frac{3}{2}}\right)\newlineNow we need to integrate this expression with respect to t from 00 to 11.
  5. Evaluating outer integral: The outer integral is: ∫01103[(1+t)32−t32]dt\int_{0}^{1}\frac{10}{3}\left[(1+t)^{\frac{3}{2}} - t^{\frac{3}{2}}\right]dt We will integrate this term by term.
  6. Final evaluation: The antiderivative of (1+t)32(1+t)^{\frac{3}{2}} with respect to tt is:\newline25⋅(1+t)52\frac{2}{5}\cdot(1+t)^{\frac{5}{2}}\newlineThe antiderivative of t32t^{\frac{3}{2}} with respect to tt is:\newline25⋅t52\frac{2}{5}\cdot t^{\frac{5}{2}}\newlineSo the integral becomes:\newline103⋅25⋅[(1+t)52−t52]\frac{10}{3}\cdot\frac{2}{5}\cdot\left[(1+t)^{\frac{5}{2}} - t^{\frac{5}{2}}\right]
  7. Calculating numerical value: We now evaluate this from t=0t=0 to t=1t=1:
    (\frac{10}{3})\cdot(\frac{2}{5})\cdot[(2)^{\frac{5}{2}} - (1)^{\frac{5}{2}}] - (\frac{10}{3})\cdot(\frac{2}{5})\cdot[(0)^{\frac{5}{2}} - (0)^{\frac{5}{2}}]\
    This simplifies to:
    \$(\frac{10}{3})\cdot(\frac{2}{5})\cdot[2^{\frac{5}{2}} - 1]
  8. Simplifying the expression: Calculating the numerical value, we get:\newline(103)⋅(25)⋅[32−1](\frac{10}{3})\cdot(\frac{2}{5})\cdot[\sqrt{32} - 1]\newline= (43)⋅[32−1](\frac{4}{3})\cdot[\sqrt{32} - 1]\newline= (43)⋅[42−1](\frac{4}{3})\cdot[4\sqrt{2} - 1]\newline= (43)⋅(42−1)(\frac{4}{3})\cdot(4\sqrt{2} - 1)
  9. Simplifying the expression: Calculating the numerical value, we get:\newline(103)⋅(25)⋅[32−1](\frac{10}{3})\cdot(\frac{2}{5})\cdot[\sqrt{32} - 1]\newline= (43)⋅[32−1](\frac{4}{3})\cdot[\sqrt{32} - 1]\newline= (43)⋅[42−1](\frac{4}{3})\cdot[4\sqrt{2} - 1]\newline= (43)⋅(42−1)(\frac{4}{3})\cdot(4\sqrt{2} - 1)Finally, we simplify the expression to get the final answer:\newline(43)⋅(42−1)(\frac{4}{3})\cdot(4\sqrt{2} - 1)\newline= (163)2−(43)(\frac{16}{3})\sqrt{2} - (\frac{4}{3})

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