document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1140897>Question 1140897</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Consider the integral 2 1[19  -  4x dt.<BR>\n" );
document.write( "(a) Using the trapezoidal rule with 4 subintervals, estimate the integral numerically. <BR>\n" );
document.write( "(b) Find the integral exactly (using calculus and showing the working) and compare<BR>\n" );
document.write( "this with your numerical result by calculating the absolute value of the error.<BR>\n" );
document.write( " </SPAN>\n" );
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document.write( "I read this as the integral from 1 to 2 of (19-4x).  [(dx..., not dt)]<br><BR>\n" );
document.write( "The interval 1 to 2 with 4 subintervals means the x values of the endpoints of the intervals are 1, 1.25, 1.5, 1.75, and 2.<br><BR>\n" );
document.write( "The function values at those endpoints are 15, 14, 13, 12, and 11.<br><BR>\n" );
document.write( "The integral by the trapezoidal rule is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F4%29%28%2815%2B14%29%2F2%2B%2814%2B13%29%2F2%2B%2813%2B12%29%2F2%2B%2812%2B11%29%2F2%29+=+%281%2F8%29%2815%2B28%2B26%2B24%2B11%29+=+104%2F8+=+13 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The indefinite integral of the function using calculus is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19x-2x%5E2 ALIGN=MIDDLE  ALT=\"19x-2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The definite integral for the interval 1 to 2 is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2819%282%29-2%282%5E2%29%29-%2819%281%29-2%281%5E2%29%29+=+%2838-8%29-%2819-2%29+=+30-17+=+13 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The \"estimate\" using the trapezoidal rule is exactly the same as the result using calculus.  This is as it should be, because the given function is linear.  For any linear function, the trapezoidal rule will give the exact integral. </SPAN>\n" );
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