Question

the graph of ( y = f(t) ) is given below:

the definite integral ( int_{0}^{6} f(t) dt ) equals
(your answer should be a number.)
view hint for question 6

Explanation:

Step1: Calculate the area of the first triangle

The first triangle has a base of \(1\) and a height of \(1\). The area of a triangle is \(A=\frac{1}{2}\times base\times height\). So, \(A_1=\frac{1}{2}\times1\times1=\frac{1}{2}\).

Step2: Calculate the area of the second triangle

The second triangle has a base of \(4\) and a height of \(3\). Using the area formula \(A = \frac{1}{2}\times base\times height\), we get \(A_2=\frac{1}{2}\times4\times3 = 6\).

Step3: Calculate the total definite - integral

The definite integral \(\int_{0}^{6}f(t)dt\) is the sum of the areas of the regions under the curve. Since the first region (from \(t = 0\) to \(t=2\)) has an area of \(\frac{1}{2}\) (but wait, re - evaluating: from \(t = 0\) to \(t = 2\), the area of the triangle below the \(t\) - axis: base \(b = 2\), height \(h=1\), area \(A_{below}=\frac{1}{2}\times2\times1 = 1\) (but with a negative sign as it is below the \(t\) - axis). From \(t=2\) to \(t = 6\), base \(b = 4\), height \(h = 3\), area \(A_{above}=\frac{1}{2}\times4\times3=6\).
The definite integral \(\int_{0}^{6}f(t)dt=- 1+6\).

Answer:

\(5\)