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

Question: Evaluate the definite integral from 1 to 4 of the vector-valued function \(F(t) = 6t^2\mathbf{i} + 8t^3\mathbf{j} + 9t^2\mathbf{k}\). Answer: The definite integral of the given vector-valued function from 1 to 4 is \(126\mathbf{i} + 510\mathbf{j} + 189\mathbf{k}\).

Step-by-step solution

Evaluate the integral of the i-component

First, we will find the integral of the i-component, which is \(6t^2\). The integral of a scalar function is computed by reversing the process of differentiation. To find the integral of \(6t^2\) with respect to t, we will use the power rule of integration which states that \(\int t^n dt = \frac{t^{n+1}}{n+1} + C\), where n is any real constant, and C is the constant of integration. Applying the power rule to \(6t^2\), we get: $$\int 6t^2 dt = 6 \cdot \frac{t^{2+1}}{2+1} = 2t^3$$

Apply the limits for the i-component

Now we will apply the definite integral limits to the evaluated i-component (1 to 4). For a definite integral with limits a and b, it can be evaluated as \(F(b) - F(a)\), where F(t) is the integration result we derived in Step 1. $$\int_{1}^{4} 6t^2 dt = 2(4^3) - 2(1^3) = 128 - 2 = 126\mathbf{i}$$

Evaluate the integral of the j-component

Next, we will find the integral of the j-component, which is \(8t^3\). Applying the power rule to \(8t^3\), we get: $$\int 8t^3 dt = 8 \cdot \frac{t^{3+1}}{3+1} = 2t^4$$

Apply the limits for the j-component

Now we will apply the definite integral limits to the evaluated j-component (1 to 4). $$\int_{1}^{4} 8t^3 dt = 2(4^4) - 2(1^4) = 512 - 2 = 510\mathbf{j}$$

Evaluate the integral of the k-component

Lastly, we will find the integral of the k-component, which is \(9t^2\). Applying the power rule to \(9t^2\), we get: $$\int 9t^2 dt = 9 \cdot \frac{t^{2+1}}{2+1} = 3t^3$$

Apply the limits for the k-component

Now we will apply the definite integral limits to the evaluated k-component (1 to 4). $$\int_{1}^{4} 9t^2 dt = 3(4^3) - 3(1^3) = 192 - 3 = 189\mathbf{k}$$

Combine the i, j, and k-components to find the final answer

Finally, we will combine the obtained results for each component to get the final answer: $$\int_{1}^{4}\left(6 t^{2} \mathbf{i}+8 t^{3} \mathbf{j}+9 t^{2} \mathbf{k}\right) d t = 126\mathbf{i} + 510\mathbf{j} + 189\mathbf{k}$$