Solve the given problems by integration. A space vehicle is...

Solve the given problems by integration. A space vehicle is launched vertically from the ground such that its velocity $v(\text { in } \mathrm{km} / \mathrm{s}$ ) is given by $$v=\left[\ln ^{2}\left(t^{3}+1\right)\right] \frac{t^{2}}{t^{3}+1}$$, where $t$ is the time (in s). Find the altitude of the vehicle after $10.0 \mathrm{s}$

Solve the given problems by integration.
A space vehicle is launched vertically from the ground such that its velocity $v(\text { in } \mathrm{km} / \mathrm{s}$ ) is given by $$v=\left[\ln ^{2}\left(t^{3}+1\right)\right] \frac{t^{2}}{t^{3}+1}$$, where $t$ is the time (in s). Find the altitude of the vehicle after $10.0 \mathrm{s}$

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes the connection between differentiation and integration. It states that if a function is continuous over an interval, its definite integral over that interval can be computed using any one of its antiderivatives. This concept allows one to evaluate the accumulation of quantities, such as displacement from a velocity function, by finding an antiderivative and evaluating it at the boundaries of the interval.

Definite Integration

Definite integration involves computing the integral of a function over a specific interval. It is used to find the net accumulation of a quantity, which could represent physical quantities like area or displacement. In many real-world applications such as kinematics, definite integration is applied to determine how a quantity changes over time by summing infinitesimal contributions.

Substitution Method

The substitution method, or u-substitution, is a technique used to simplify integrals by changing variables. This method is particularly useful when dealing with composite functions, where a substitution can transform the integrand into a form that is easier to integrate. It effectively reverses the chain rule used in differentiation, enabling the evaluation of more complex integrals.

Integration in Kinematics

Integration in kinematics is used to determine quantities like displacement, velocity, and acceleration by integrating one with respect to another. For instance, integrating a velocity function with respect to time provides the displacement (or change in position) over a given time interval. This approach is essential in understanding the dynamics of moving objects and is widely applicable in physics and engineering problems.

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