Question Video: Finding the Area of the Region between a Curve and Two Straight Lines | Nagwa Question Video: Finding the Area of the Region between a Curve and Two Straight Lines | Nagwa

# Question Video: Finding the Area of the Region between a Curve and Two Straight Lines Mathematics • Third Year of Secondary School

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Find the area of the shaded region.

01:49

### Video Transcript

Find the area of the shaded region.

Letโs use the given figure to first determine the equations of each of the lines that bound this region. First, we have the curve ๐ฆ equals three ๐ฅ squared plus four ๐ฅ minus two. We also have the vertical lines ๐ฅ equals one and ๐ฅ equals two. But rather than our region being bounded by the ๐ฅ- or ๐ฆ-axis, the final line which bounds this region is the line ๐ฆ equals five.

Now if we were just looking to find the area bounded by the quadratic graph, the lines ๐ฅ equals one and ๐ฅ equals two on the ๐ฅ-axis, we could evaluate the definite integral from one to two of three ๐ฅ squared plus four ๐ฅ minus two with respect to ๐ฅ. Notice though that this would include the area of the rectangle below the line ๐ฆ equals five. And so to find the area of just the pink region, weโd need to subtract the area of this rectangle from our integral.

The area of this rectangle isnโt tricky to find. It has a vertical height of five units and a width of one unit. Thatโs two minus one. So its area is just five multiplied by one, or five. This then gives us one method that we can use. We evaluate the definite integral to find the area all the way down to the ๐ฅ-axis and then subtract the area of the orange rectangle. Working through the integral would give an answer of six square units.

But there is actually another way that we could answer this question. The area below the line ๐ฆ equals five can also be found using integration. Itโs equal to the integral from one to two of five with respect to ๐ฅ. And using the linear property of integration, we could then express this as a single integral. If you were to form this integral, it would give the same result.

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