Question

Evaluate the line integral, where C is the given curve. \int y3\ ds,\ C\div x=t3,\ y=t,\ 0?\ t?\ 3

Integrals
ANSWERED
asked 2021-06-13
Evaluate the line integral, where C is the given curve. \(\int y3\ ds,\ C\div x=t3,\ y=t,\ 0?\ t?\ 3\)

Answers (1)

2021-06-14

Step 1
\(f(x,\ y)=y^{3}\)
\(r(t)=\langle t^{3},\ t \rangle\)
\(r'(t)=\langle3t^{2},\ 1\rangle\)
\(ds=|r'(t)|dt\)
\(ds=\sqrt{(3t^{2})^{2}+1^{2}}dt\)
\(ds=\sqrt{(9t^{4})+1}dt\)
Integral \(y^{3}ds\)
Integral \(=\int_{[0\ to\ 3]}f(r(t))\cdot ds\)
\(=\int_{[0\ to\ 3]}t^{3}\cdot\sqrt{[(9t^{4})+1]}dt\)
Let \((9t^{4})+1=u\)
\(\Rightarrow9\times4t^{3}dt+0=du\Rightarrow t^{3}dt=\frac{dt}{36}\)
\(\Rightarrow\int^{3}\cdot\sqrt{(9t^{4})+1}dt\)
\(\Rightarrow\int\sqrt{u}\frac{du}{36}\)
\(\Rightarrow\left(\frac{1}{36}\right)\frac{u^{\frac{3}{2}}}{(\frac{3}{2})}+c\)
\(\Rightarrow\left(\frac{1}{54}\right)u^{\frac{3}{2}}+c\)
\(\Rightarrow\left(\frac{1}{54}\right)[(9t^{4})+1]^{\frac{3}{2}}+c\)
\(\int_{0}^{3}t^{3}\cdot\sqrt{(9t^{4})+1]}dt=\int_{0}^{3}\left(\frac{1}{54}\right)[(9t^{4})+1]^{\frac{3}{2}}+c\)
\(\Rightarrow\left(\frac{1}{54}\right)[(9\times3^{4})+1]^{\frac{3}{2}}+c-\left(\frac{1}{54}\right)[(9\times0^{4})+1]^{\frac{3}{2}}-c\)
\(\Rightarrow\left(\frac{1}{54}\right)[730]^{\frac{3}{2}}-\left(\frac{1}{54}\right)\)
\(\Rightarrow\left(\frac{1}{54}\right)[730^{\frac{3}{2}}-1]\)
\(\Rightarrow365.23\)

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