Integrals of Vector-Valued Functions
Solved Problems
Example 7.
Find \(\mathbf{R}\left( t \right)\) if \[\mathbf{R}^\prime\left( t \right) = \left\langle {\sin \frac{t}{3},\cos \frac{t}{2}} \right\rangle \] and \(\mathbf{R}\left( \pi \right) = \left\langle {\frac{1}{2},\frac{1}{2}} \right\rangle .\)
Solution.
Integrating the vector function yields:
\[\mathbf{R}\left( t \right) = \int {\left\langle {\sin \frac{t}{3},\cos \frac{t}{2}} \right\rangle dt} = \left\langle {\int {\sin \frac{t}{3}dt} ,\int {\cos \frac{t}{2}dt} } \right\rangle = \left\langle { - 3\cos \frac{t}{3},2\sin \frac{t}{2}} \right\rangle + \mathbf{C}.\]
We find the vector \(\mathbf{C} = \left\langle {{C_1},{C_2}} \right\rangle \) from the initial condition \(\mathbf{R}\left( \pi \right) = \left\langle {\frac{1}{2},\frac{1}{2}} \right\rangle :\)
\[\mathbf{R}\left( \pi \right) = \left\langle { - 3\cos \frac{\pi }{3},2\sin \frac{\pi }{2}} \right\rangle + \mathbf{C} = \left\langle { - 3 \cdot \frac{1}{2},2 \cdot 1} \right\rangle + \mathbf{C} = \left\langle { - \frac{3}{2},2} \right\rangle + \mathbf{C} = \left\langle {\frac{1}{2},\frac{1}{2}} \right\rangle .\]
Then
\[\mathbf{C} = \left\langle {\frac{1}{2},\frac{1}{2}} \right\rangle - \left\langle { - \frac{3}{2},2} \right\rangle = \left\langle {2, - \frac{3}{2}} \right\rangle .\]
The final answer is
\[\mathbf{R}\left( t \right) = \left\langle { - 3\cos \frac{t}{3},2\sin \frac{t}{2}} \right\rangle + \left\langle {2, - \frac{3}{2}} \right\rangle .\]
Example 8.
Compute the integral \[\int\limits_0^1 {\left\langle {\frac{{2t}}{{1 + {t^2}}},\frac{2}{{1 + {t^2}}}} \right\rangle dt}.\]
Solution.
Integrating component-by-component, we can write:
\[\int\limits_0^1 {\left\langle {\frac{{2t}}{{1 + {t^2}}},\frac{2}{{1 + {t^2}}}} \right\rangle dt} = \left\langle {\int\limits_0^1 {\frac{{2tdt}}{{1 + {t^2}}}} ,\int\limits_0^1 {\frac{{2dt}}{{1 + {t^2}}}} } \right\rangle .\]
We evaluate each of the integrals separately. To find the first integral, we make the substitution \(u = 1 + {t^2},\) \(du = 2tdt.\) Then
\[\int {\frac{{2tdt}}{{1 + {t^2}}}} = \int {\frac{{du}}{u}} = \ln \left| u \right| = \ln \left( {1 + {t^2}} \right),\]
and
\[\int\limits_0^1 {\frac{{2tdt}}{{1 + {t^2}}}} = \left. {\ln \left( {1 + {t^2}} \right)} \right|_0^1 = \ln 2 - \ln 0 = \ln 2.\]
Calculate the second integral:
\[\int\limits_0^1 {\frac{{2dt}}{{1 + {t^2}}}} = \left. {2\arctan t} \right|_0^1 = 2\left( {\arctan 1 - \arctan 0} \right) = 2\left( {\frac{\pi }{4} - 0} \right) = \frac{\pi }{2}.\]
Thus, the initial integral is represented by the vector
\[\int\limits_0^1 {\left\langle {\frac{{2t}}{{1 + {t^2}}},\frac{2}{{1 + {t^2}}}} \right\rangle dt} = \left\langle {\ln 2,\pi } \right\rangle .\]
Example 9.
Compute the integral \[\int\limits_0^{\frac{\pi }{6}} {\left\langle {2\cos t,\sin 2t} \right\rangle dt}.\]
Solution.
By integrating componentwise, we obtain:
\[\int\limits_0^{\frac{\pi }{6}} {\left\langle {2\cos t,\sin 2t} \right\rangle dt} = \left\langle {\int\limits_0^{\frac{\pi }{6}} {2\cos tdt} ,\int\limits_0^{\frac{\pi }{6}} {\sin 2tdt} } \right\rangle = \left\langle {\left. {2\sin t} \right|_0^{\frac{\pi }{6}}, - \left. {\frac{{\cos 2t}}{2}} \right|_0^{\frac{\pi }{6}}} \right\rangle = \left\langle {2\left( {\frac{1}{2} - 0} \right), - \frac{1}{2}\left( {\frac{1}{2} - 1} \right)} \right\rangle = \left\langle {1,\frac{1}{4}} \right\rangle .\]
Example 10.
A particle starts moving from the origin with the velocity \[\mathbf{v}\left( t \right) = \left\langle {4t,3{t^2} - 1} \right\rangle,\] where \(t\) is measured in seconds and the components are in meters per second. Determine the displacement \(\left| \mathbf{r} \right|\) of the particle in \(t = 2\) seconds.
Solution.
Determine the position of the particle in \(t = 2\) seconds by integrating the velocity vector:
\[\mathbf{r}\left( t \right) = \int\limits_0^2 {\mathbf{v}\left( t \right)dt} = \int\limits_0^2 {\left\langle {4t,3{t^2} - 1} \right\rangle dt} = \left\langle {\int\limits_0^2 {4tdt} ,\int\limits_0^2 {\left( {3{t^2} - 1} \right)dt} } \right\rangle = \left\langle {\left. {2{t^2}} \right|_0^2,\left. {{t^3} - t} \right|_0^2} \right\rangle = \left\langle {8,8 - 2} \right\rangle = \left\langle {8,6} \right\rangle .\]
Hence, the displacement of the particle is equal
\[\left| {\mathbf{r}\left( t \right)} \right| = \sqrt {{8^2} + {6^2}} = 10\,\text{m}.\]
Example 11.
A particle starts moving from the origin with the acceleration \[\mathbf{a}\left( t \right) = \left\langle {6t,4} \right\rangle,\] where \(t\) is measured in seconds and the components are in meters per seconds squared. Determine the displacement \(\left| \mathbf{r} \right|\) of the particle in \(t = 1\) seconds if \(\mathbf{v}\left( 0 \right) = \left\langle {2,2} \right\rangle .\)
Solution.
First we integrate the acceleration vector to obtain the velocity vector:
\[\mathbf{v}\left( t \right) = \int {\mathbf{a}\left( t \right)dt} = \int {\left\langle {6t,4} \right\rangle dt} = \left\langle {3{t^2} + {k_1},4t + {k_2}} \right\rangle ,\]
where \(\mathbf{K} = \left\langle {{k_1},{k_2}} \right\rangle \) is a constant vector depending on the initial conditions. We determine this vector from the initial condition \(\mathbf{v}\left( 0 \right) = \left\langle {2,2} \right\rangle :\)
\[\mathbf{v}\left( 0 \right) = \left\langle {3{t^2} + {k_1},4t + {k_2}} \right\rangle = \left\langle {0 + {k_1},0 + {k_2}} \right\rangle = \left\langle {{k_1},{k_2}} \right\rangle = \left\langle {2,2} \right\rangle .\]
Hence, the particle's velocity is defined by the vector
\[\mathbf{v}\left( t \right) = \left\langle {3{t^2} + 2,4t + 2} \right\rangle .\]
Now we integrate the velocity vector to get the position vector:
\[\mathbf{r}\left( t \right) = \int {\mathbf{v}\left( t \right)dt} = \int {\left\langle {3{t^2} + 2,4t + 2} \right\rangle dt} = \left\langle {{t^3} + 2t + {m_1},\,2{t^2} + 2t + {m_2}} \right\rangle .\]
The constant vector \(\mathbf{M} = \left\langle {{m_1},{m_2}} \right\rangle \) also depends on the initial condition. As \(\mathbf{r}\left( 0 \right) = \left\langle {0,0} \right\rangle ,\) we have \(\mathbf{M} = \left\langle {{0},{0}} \right\rangle .\)
So the position vector is given by
\[\mathbf{r}\left( t \right) = \left\langle {{t^3} + 2t,2{t^2} + 2t} \right\rangle .\]
Calculate the coordinates of the particle at \(t = 1:\)
\[\mathbf{r}\left( {t = 1} \right) = \left\langle {{1^3} + 2 \cdot 1,2 \cdot {1^2} + 2 \cdot 1} \right\rangle = \left\langle {3,4} \right\rangle .\]
Then the displacement of the particle is equal
\[\left| {\mathbf{r}\left( {t = 1} \right)} \right| = \sqrt {{3^2} + {4^2}} = 5\,\text{m}.\]
