6.6. Example Problems

Example 1: Find the Laplace Transform of

$g(t) = \int_0^t \sin(a\tau) d\tau$

Solution: Let f(t)=sin(at), then

$F(s)=L\{\sin(at)\}=\frac{a}{s^2 + a^2}$

Using the theorem,

$L\{\int_0^t \sin(a\tau) d\tau\} = \frac{F(s)}{s} = \frac{a}{s(s^2 + a^2)}$

In this example, we need to find the Laplace transform of the integral of the sine function. We denote the integral as g(t) which represents the accumulated area under the curve defined by sin(aτ). According to our initial setup, the Laplace Transform of sin(at) gives us F(s). Applying the theorem allows us to express the Laplace Transform of the integral in terms of F(s). By substituting F(s) into our equation, we derive the final result that shows how the Laplace Transform of the integral can be simplified.

Think about filling a tank with water where the rate of water flowing in follows a sine wave pattern. The integral represents the total amount of water in the tank at time t. By calculating the Laplace Transform, we can analyze how the tank behaves over time, despite the changing rate of flow.

Example 2: Find $L\{ \int_0^t e^{2\tau} d\tau \}$

Solution: Let f(t)=e^{2t} \Rightarrow F(s)= \frac{1}{s-2}, s>2

So,

$L\{\int_0^t e^{2\tau} d\tau \} = \frac{1}{s(s-2)}$

In this second example, we want to find the Laplace Transform of another type of integral, specifically one involving an exponential function. We start by expressing f(t) as e^(2t) and then compute its Laplace Transform to obtain F(s). Using the given theorem again allows us to relate the Laplace Transform of the integral directly to the calculated F(s). This demonstrates how different functions can be transformed similarly using the same theoretical approach.

Imagine a bank account where money is continuously added at an exponential rate (representing e^(2τ)). The integral calculates the total amount over time, which might represent your total balance at any given moment. The Laplace Transform helps us model this accumulation in a useful mathematical form, allowing us to predict future balances based on current rates.

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