Question

Evaluate the following definite integrals. $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t$$

Short answer

Given: $$f(t) = \langle 6t^2 \mathbf{i}, 8t^3\mathbf{j}, 9t^2\mathbf{k} \rangle$$ Limits of integration: 1 to 4 Solution: Evaluate the definite integral for each component: $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t = \langle126\mathbf{i}, 510\mathbf{j}, 189\mathbf{k}\rangle$$

Step-by-step solution

Identify the vector components and the limits of integration

The vector-valued function is given by \( \langle 6t^2 \mathbf{i}, 8t^3 \mathbf{j}, 9t^2 \mathbf{k} \rangle\), represented by scalar functions \(6t^2\), \(8t^3\), and \(9t^2\) for \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) respectively. The limits of integration are from 1 to 4.

Evaluate the definite integrals of each component

Calculate the definite integrals for each of the scalar functions: For the \(\mathbf{i}\) component: $$\int_{1}^{4}6t^2 dt$$ For the \(\mathbf{j}\) component: $$\int_{1}^{4}8t^3 dt$$ For the \(\mathbf{k}\) component: $$\int_{1}^{4}9t^2 dt$$

Calculate the antiderivatives of the scalar functions

Find the antiderivative F(x) for each scalar function: For the \(\mathbf{i}\) component: $$F(t) = 2t^3$$ For the \(\mathbf{j}\) component: $$F(t) = 2t^4$$ For the \(\mathbf{k}\) component: $$F(t) = 3t^3$$

Use the limits of integration to evaluate the antiderivatives

Now, we will use the limits of integration to evaluate the antiderivative functions at the upper and lower bounds and calculate their respective differences. For the \(\mathbf{i}\) component: $$2(4)^3 - 2(1)^3 = 128 - 2 = 126$$ For the \(\mathbf{j}\) component: $$2(4)^4 - 2(1)^4 = 512 - 2 = 510$$ For the \(\mathbf{k}\) component: $$3(4)^3 - 3(1)^3 = 192 - 3 = 189$$

Combine the results into a vector

Combine the results from steps 3 and 4 to form the resultant vector: $$\langle126\mathbf{i}, 510\mathbf{j}, 189\mathbf{k}\rangle$$ So, the definite integral is: $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t = \langle126\mathbf{i}, 510\mathbf{j}, 189\mathbf{k}\rangle$$