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Greeley [361]
3 years ago
14

Use the graph of f to calculate

ormula1" title="\int\limits^4_0 {f(t)} \, dt" alt="\int\limits^4_0 {f(t)} \, dt" align="absmiddle" class="latex-formula">. PLEASE HELP!

Mathematics
1 answer:
frozen [14]3 years ago
3 0

You should already be aware that the definite integral of f(t) is equivalent to the signed area under the curve of

\displaystyle\int_0^1f(t)\,\mathrm dt

corresponds the area of the triangle on the left, and

\displaystyle\int_1^4f(t)\,\mathrm dt

is the area of the trapezoid in the middle. You should know how to compute the area of these shapes.

We end up with

\displaystyle\int_0^4f(t)\,\mathrm dt=-\frac{1\times2}2+\frac{(1+3)\times2}2=3

since the triangle has height 2 and base 1, and the trapezoid has height 2 with two base lengths 1 and 3.

Edit: The area under the curve is highlighted red in the attachment below. It's the area between the curve and the horizontal axis, but it's also signed, meaning the area <em>below</em> the axis is negative, and the area <em>above</em> the axis is positive.

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