Advanced Math Archive: Questions from July 31, 2022
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2 Determine the following integrals \( 2.1 \int_{0}^{\frac{\pi}{2}} x^{2} \sin x d x \). \( 2.2 \int \frac{14 x^{2}-7 x-3}{\left(x^{2}-1\right)(2 x-1)} d x \). \( 2.3 \int \frac{3 x-6}{x^{2}+10 x+28}1 answer -
Find the domain of the function \( f(x, y)=\sqrt{5 x^{2}-5 y+1} \). The set of all ordered pairs \( (x, y) \) for which: A) \( y\frac{1+5 x^{2}}{5} \) (D) \( y \geq-\frac{1+5 x^{2}}{5} \) (E) \( y \le1 answer -
Solve the DE'S:- \[ \text { 1. } 3 t^{2}+4 y^{3} y^{\prime}-1+y^{\prime}=0 \] \( y^{\prime}=t^{2}-y^{3} \)1 answer -
Solve the IVP:- II \( \left(6 x^{5}-x y\right)+\left(-x^{2}+x y^{2}\right) y^{\prime}=0 \quad, y(0)=1 \)1 answer -
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Let: \[ U=\{q, r, s, t, u, v, w, x, y, z\} \] \[ A=\{q, s, u, w, y\} \] \[ B=\{q, s, y, z\} \] \[ C=\{v, w, x, y, z\} \] List the elements in the set \( (A \cap B) \) ' \( \{r, t, u, v, w, x, z\} \) \3 answers -
8. (10 points) Evaluate \( \iiint_{1}\left(x^{2}+y^{2}+z^{2}\right) d V \) where \( \Gamma=\left\{(x, y, z) \mid 0 \leq x \leq \sqrt{1-y^{2}}, 0 \leq\right. \) \( \left.y \leq 1, \sqrt{x^{2}+y^{2}} \l1 answer